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Class TU 5G 1 3 


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Copyright N°_ 

COPYRIGHT DEPOSIT; 










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THE CARPENTERS’ 

Steel Square, 

AND ITS USES 


BEING A DESCRIPTION OF THE SQUARE, AND ITS USES IN OBTAI IHG THE 
LENGTHS AND BEVELS OF ALL KINDS OF 


RAFTERS, HIPS, GROINS, BRACES, BRACKETS, PURLINS, 
COLLAR BEAMS, AND JACK=RAFTERS; 

ALSO ITS APPLICATION IN OBTAINING THE BEVELS AND CUTS FOR 
HOPPERS, SPRING MOULDINGS, OCTAGONS, STAIRS, 
DIMINISHED STILES, ETC. 

ILLUSTRATED BY NEARLY ONE HUNDRED ENGRAVINGS. 

BY 

FRED. T. HODGSON, 

if - ii 

Editor of the “Builder and Woodworker." 

Fourth Edition. Revised and Greatly Enlarged. 

o 

> > > 

> » 

New YOrk: 


THE INDUSTRIAL PUBLICATION! COMPANY, 

1GGG 



I 



LIBRARY of CONGRESS 
Two Cooies Received 

JUN 27 1906 





Copyright Secured 
1880 , 1883 , 1906 . 
By J OHN PHIN, 



f' 

$ 




PREFACE TO SECOND EDITION, 


The rapid disposal of the first edition of the “Steel 
Square and Its Uses,” has rendered it incumbent for the 
publisher to issue a second and larger edition; and recog¬ 
nizing this condition, in connection with the fact that the 
work has met with more than a passing favor from those 
who make daily use of the Steel Square, it has been 
deemed necessary to make the present edition more 
useful by adding a number of solutions of mechanical 
problems by aid of the instrument, and other matters 
that will render the work more valuable to the operative 
mechanic. 

The Author has reason to, and does, feel pleased at the 
appreciation the working mechanics of this country have 
evinced for this work; and is assured, by the numerous 
letters, and other indications of good feeling he has re¬ 
ceived on all hands, that the present enlargement of the 
work has not been made unnecessarily or too soon. 

Feeling confident that the additions to the present edi¬ 
tion will commend themselves to the toiling thousands 
who have daily use for the “Steel Square,” the publishers 
send the enlarged work out to the public with a knowl¬ 
edge that it will be welcomed by those who are most 
interested in the subject of which it treats. 


New York, Jan. 1, 1883 










■*> 











PREFACE 


\ 


Some time ago, the author of this little work contributed a series 
of papers on the Steel Square and Its Uses, to the American 
Builder, and since their appearance, he has received hundreds of 
letters from as many persons residing in various parts of the 
United States, Canada, Australia and New Zealand, in which the 
writers requested him to publish the papers in book form. Partly 
in compliance with these requests, and partly at the solicitation 
of personal friends, together with a knowledge that a ch '•ap but 
thorough work of the kind, would be of service to all persons who 
have occasion to use a steel square, he has consented, with the aid 
of the present enterprising publishers, to issue the work as now 
offered. 

♦ 

It is only of late years that American workmen have begun 
fully to understand the capabilities of the steel square; and even 
now, only a few of the best workmen have any idea of what can be 
accomplished with it when in skilful hands. 

It is not claimed that the rules and methods shown in this little 
work are either new or original; they have been known to advanced 
workmen for many years past; but it is claimed that they have 
never before been brought together and put in so handy a shape as 



PREFACE. 


Vi 

In the present book; and it is further claimed that many of the 
rules herein illustrated and explained, have never appeared in 
print previous to the publication of the papers on the subject in the 
magazine referred to above. 

Should this little volume prove of service to the man who toils 
with axe, saw and plane, for his daily bread, and profitable to the 
publishers who risk their money on its publication, it will have 
fulfilled its mission, as designed by 

THE AUTHOR 


New York, 1880 ., 


CONTENTS 


PART I., 

PAGE 

Preliminary -------- 9 

Historical and Descriptive • 13 

Description of the Square - - - - - - 16 

Board, Plank and Scantling Measure 19 

Brace Buie - - - - - - - -21 

Octagonal Scale ------- 22 

Fence for Square - - - -.- - -24 

Application of Square ------ 25 

To lay out Rafters - - - - - - -28 

Hip Rafters, Cripples, etc. - - - - 33 

Backing for Hips - - - - - - -35 

Stairs and Strings ------- 3y 

Miscellaneous Rules - - - - - - -42 

Measurement - -- ...-43 

Proportion of Circles - - - - - - -46 

Centering Circles - -- -- -- 47 

How to Describe an Ellipse - - - - - - 48 

How to Describe a Parabola ----- 49 

Bevels for Hoppers - - - ' - - - -50 

Bisecting Circles - - - - - - 51 

Cutting Spring Moulding® - - - - - -54 



CONTENTS. 


••• 

vrn 

PART II. 

PAGK 

Theoretical Rafters - - . - - - - 57 

Bevels and Lengths for Hips, Jacks and Purlins 58 

Divisions of Widths - - - - - - - 60 

Eisection of Angles ------ 60 

Diminishing Stiles - - - - - - - 01 

PART III. 

• 

Octagons ---------62 

To Find the Diagonal of a Square - - - 66 

Polygons - ...... 67 

Circles -------- 68 

To Lay out Angles - - - - - - 70 

Bevels of Hoppers ----- - -70 

Widths of Sides and Ends of Hoppers - - - - 71 

Corner Pieces for Hoppers ----- 73 

Roofing - - - - - - - -73 

Lengths and Bevels of Hip-Rafters 73 

Backing of Hips - • - - - - - - 75 

Irregular Hip Roofs ------- 75 

Trusses --------- 77 

Diameter of Circles ------- 78 

Cutting Equal Mitres - - - - - - - 80 

Theoretical Lengths ------ 81 

PART TV'. 

Miscellaneous Rules and Memoranda - - - - 83 

Hip Roofs. 83-89 

Curved Hip Rafters 91 

Hopper Angles - 92 

Splayed Gothic Heads - - - - - - 95 




THE CARPENTERS’ STEEL SQUARE, 

AND ITS USES. 


PART r. 

Preliminary. —There is nothing of more importance to 
a young man who is learning the business of house-joinery 
and carpentry, than that he should make himself thoroughly 
conversant with the capabilities of the tools he employs. It 
may be that, in some of the rules shown in this work, the 
result could be attained much readier with other aids than 
the square ; but the progressive mechanic will not rest satis¬ 
fied with one method of performing operations when others 
are within his reach. 

* In the hand of the intelligent mechanic the square be¬ 
comes a simple calculating machine of the most wonderful 
capacity, and by it he solves problems of the kinds continu¬ 
ally arising m xechanical work, which by the ordinary 
methods are more difficult to perform. 

The great improvement which the arts and manufactures 
have attained within the last fifty years, renders it essential 
that every person engaged therein should use his utmost 
exertions to obtain a perfect knowledge of the trade he 



IO 


THE STEEL SQUARE 


professes to follow. It is not enough, nowadays, tor a per¬ 
son to have attained the character of a good workman; 
that phrase implies that quantum of excellence, which con¬ 
sists in working correctly and neatly, under the directions 
of others. The workman of to-day, to excel, must under¬ 
stand the principles of his trade, and be able to apply them 
correctly in practice. Such an one has a decided advan¬ 
tage over his fellow-workman; and if to his superior know¬ 
ledge he possesses a steady manner, and industrious habits, 
his efforts cannot fail of being rewarded. 

It is no sin not to know much, though it is a great one 
not to know all we can, and put it all to good use. Yet, 
how few mechanics there are who will know all they can ? 
Men apply for employment daily who claim to be finished 
mechanics, and profess to be conversant with all the ins 
and outs of their craft, and who are noways backward in 
demanding the highest wages going, who, when tested, are 
found wanting in knowledge of the simplest formulas of 
their trade. They may, perhaps, be able to perform a good 
job of work after it is laid out for them by a more compe-. 
tent hand; they may have a partial knowledge of the uses 
and application of their tools; but, generally, their know¬ 
ledge ends here. Yet some of these men have worked at 
this trade or that for a third of a century, and are to all 
appearances, satisfied with the little they learned when they 
were apprentices. True, mechanical knowledge was not 
always so easily obtained as at present, for nearly all works 
on the constructive arts were written by professional archi¬ 
tects, engineers, and designers, and however unexception¬ 
able in other respects, they were generally couched in such 
language, technical and mathematical, as to be perfectly 


AND ITS USES. 


11 

unintelligible to the majority of workmen; and instead of 
acting as aids to the ordinary inquirer, they enveloped in 
mystery the simplest solutions of every-day problems, dis¬ 
couraging nine-tenths of workmen on the very threshold of 
inquiry, and causing them to abandon further efforts to 
master the intricacies of their respective trades. 

Of late years, a number of books have been published, 
in which the authors and compilers have made commend¬ 
able efforts to simplify matters pertaining to the arts of car¬ 
pentry and joinery, and the mechanic of to-day has not 
the difficulties of his predecessors to contend with. The 
workman of old could excuse his ignorance of the higher 
branches of his trade, by saying that he had no means of 
acquiring a knowledge of them. Books were beyond his 
reach, and trade secrets were guarded so jealously, that only 
a limited few were allowed to know them, and unless he was 
made of better stuff than the most of his fellow-workmen, 
he was forced to plod on in the same groove all his days. 

Not so with the mechanic of to-day; if he is not well 
up in all the minutse of his trade, he has but himself to 
blame, for although there is no royal road to knowledge, 
there are hundreds of open ways to obtain it; and the 
young mechanic who does not avail himself of one or other 
of these ways to enrich his mind, must lack energy, or be 
altogether indifferent about his trade, and may be put down 
as one who will never make a workman. 

I have thought that it would not be out of place to pre¬ 
face this work on the “ Steel Square,” with the foregoing 
remarks, in the hope that they may stimulate the young 
mechanic, and urge him forward to conquer what at best 
are only imaginary difficulties. A willing heart and a 


12 


THE STEEL SQUARE 


clear head will most assuredly win honorable distinction in 
any trade, if they are only properly used. Indeed, during 
in experience of many years in the employment and su- 
)erintenaence of mechanics of every grade, from the green 
“ wood-haggler ” to the finished and accomplished work¬ 
man, I have invariably discovered that the finished work¬ 
man was the result of persistent study and application, and 
not, as is popularly supposed, a natural or spontaneous 
production. It is true that some men possess greater 
natural mechanical abilities than others, and consequently 
a greater aptitude in grasping the principles that underlie 
the constructive arts; but, as a rule, such men are not 
reliable; they may be expert, equal to any mechanical 
emergency, and quick at mastering details, but they are 
seldom thorough, and never reliable where long sustained 
efforts are required. 

The mechanic who reaches a fair degree of perfection by 
experience, study and application, is the man who rises to 
the surface, and whose steadiness and trustworthiness force 
themselves on the notice of employers and superinten¬ 
dents. I have said this in order to give encouragement 
to those young mechanics who find it up-hill work to 
master the intricacies of the various arts they are engaged 
in, for they may rest assured that in the end work and 
application will be sure to win; and I am certain that a 
thorough study of the steel square and its capabilities 
will do more than anything else to aid the young work¬ 
man in mastering many of ;he mechanical difficulties that 
will confront him from time to time in his daily occupation. 

It must not be supposed that the work here presented 
exhausts the subject. The enterprising mechanic will find 


AND ITS USES. 


*3 


opportunity for using the square in the solution of many 
problems that will crop up during his daily work, and the 
principles herein laid down will aid very much towards 
correct solutions. In framing roofs, bridges, trestle-work, 
and constructions of timber, the Steel Square is a necessity 
to the American carpenter; but only a few of the more in¬ 
telligent workmen ever use it for other purposes than to 
make measurements, lay off the mortices and tenons, and 
square over the various joints. Now, in framing bevel 
work of any description, the square may be used with 
great advantage and profit. Posts, girts, braces, and struts 
of every imaginable kind may be laid out by this wonderful 
instrument, if the operator will only study the plans with a 
view of making use of his square for obtaining the various 
bevels, lengths and cuts required to complete the work in 
hand. Tapering structures—the most difficult the framer 
meets with—do not contain a single bevel or length that 
can not be found by the square when properly applied, and 
it is this fact I wish to impress on my readers, for it 
would be impossible, in this work, to give every possible 
application of the square to work of this kind. I have, 
therefore, only given such examples as will enable any one 
to apply some one of them to any work in hand. 

The Square—Historical and Descriptive. —Doubtless, 
in the early ages of mankind, when solid structures be¬ 
came a necessity, the want of an instrument similar to the 
square must have been felt at every “turn and corner,” 
and there can be no question about one having been used— 
rude and imperfect perhaps—in erecting the first square 
or rectangular building that was ever built on this earth. 


«4 


THE STEEL SQUARE 


The Greeks, who were an inventive people, and wlw 
were apt to ascribe to themselves more credit than was 
really their due, in the way of inventions and discoveries, 
lay claim to be the inventors of the instrument. Pliny 
says that Theodorus, a Greek of Samos, invented the 
square and level. Theodorus was an artist of some note, 
but it is evident that the square and level, in some form 
or other, were used long before his time, even in his own 
country, for some of the finest temples in Athens and other 
Grecian cities, had been built long before his time; and 
the Pyramids of Egypt were hoary with age when he was 
in swaddling cloths. Indeed, the “ square,” as a construc¬ 
tive tool, must of necessity have found a place in the 
“ kit ” of the earliest builders. Evidences of its presence 
have been found in the ruins of pre-historic nations, and 
are abundant in the remains of ancient Petra, Nineveh, 
Babylon, Etruria, and India. South American ruins of 
great antiquity in Brazil, Peru, and other places, show that 
the unknown races that once inhabited the South American 
Continent, were familiar with many of the uses of the 
square. Egypt, however, that cradle of all the arts, fur¬ 
nishes us with the most numerous, and, perhaps, the most 
ancient evidences of the use of the square; paintings and 
inscriptions on the rock-cut tombs, the temples, and other 
works, showing its use and application, are plentiful. In 
one instance, a whole “ kit ” of tools was found in a tomb 
at Thebes, which consisted of mallets, hammers, bronze 
nails, small tools, drills, hatchets, adzes, squares, chisels, 
etc.; one bronze saw and one adze have the name of 
Thothmes III., of the -i8th dynasty, stamped on their 
blades, showing that they were made nearly 3,500 years 


AND ITS USES. 


*s 

ago. The constructive and decorative arts at that time 
were in their zenith in Egypt, and must have taken at least 
1,000 years to reach that stage. Consequently, the 
square must have been used by workmen of that country, 
at least, four thousand years ago. 

The British Museum contains many tools of pre-historic 
origin, and the square is not the least of them. Hercu¬ 
laneum and Pompeii contribute evidences of the importance 
of this useful tool. On some of the paintings recently dis¬ 
covered in those cities, the different artisans can be seen at 
home in their own workshops, with their work-benches, 
saw-horses, tools, and surroundings, much about the same 
as we would find a small carpenter shop of to-day, where 
all the work is done by hand; the only difference being a 
change in the form of some of the tools, which, in some 
instances, had been better left as these old workmen de¬ 
vised them. 

% 

It can make no difference, however, to the modern 
workman, as to when or where the square was first used; 
suffice to know, that, at present, we have squares im¬ 
mensely superior to anything known to the ancients, and 
it may be added, that so perfect has the machinery for 
the manufacture of steel squares become, that a defective 
tool is now the exception. Of course this relates to the 
products of manufacturers of repute, and not to the cheap 
squares, or to those said to be “ first-class,” that were made 
ten or fifteen years ago. The tool we recommend else¬ 
where is the best made, both as to quality of material, 
accuracy of workmanship, and amount cf useful matter on 
its faces. 


16 


THE STEEL SQUARE 


Description of the Square.— In the foregoing sketch I 

have given a few hints as to the kind of square to purchase 
when it is necessary to buy; in many cases, however, this 
book will find its way into the hands of mechanics and 
others, who will have old and favorite squares in their 
chests or workshops, and who will not care to dispose of 
a “ well-tried friend ” for the purpose of filling its place 
with another, simply because I have recommended it. 1 o 
these workmen I would say that I do not advise a change, 
provided the old square is true, and the inches and sub¬ 
divisions are properly and accurately defined. I wish it 
distinctly understood that old squares, if true, and marked 
with inches and sub-divisions of inches, will perform nearly 
every solution presented in this book. 

The lines and figures formed on the squares of different 
make, sometimes vary, both as to their position on the 
square, and their mode of application, but a thorough 
understanding of the application of the scales and lines 
shown on any first-class tool, will enable the student to 
Comprehend the use of the lines and figures exhibited on 
other first-class squares. 

To insure good results, it is necessary to be careful in 
the selection of the tool. The blade of the square should 
be 24 inches long, and two inches wide, and the tongue 
from 14 to 18 inches long and 1 x / 2 inches wide. The 
tongue should be exactly at right angles with the blade, 
or in other words the “square” should be perfectly square. 

To test this question, gee a board, about 12 or 14 inches 
wide, and four feet long, dress it on one side, and true up one 
edge as near straight as it is possible to make it. Lay the 
board on the bench, with the dressed side up, and the 



AND ITS USES. 


*7 


trued edge towards you, then apply the square, with the 
blade to the left, and mark across the prepared board with 
a penknife blade, pressing close against the edge of the 
tongue ; this process done to your satisfaction, reverse the 
square, and move it until the tongue is close up to the 
knife mark; if you find that the edge of the tongue and 
mark coincide, it is proof that the tool is correct enough 
for your purposes. 

This, of course, relates to the inside edge of the blade, 
and the outside edge of the tongue. If these edges should 
not be straight, or should not prove perfectly true, they 
should be filed or ground until they are straight and true. 
The outside edge of the blade should also be “ trued ” up 
and made exactly parallel with the inside edge, if such 
is required. The same process shuuld be gone through 
on the tongue. As a rule, squares made by firms of 
repute are perfect, and require no adjusting; nevertheless, 
it is well to make a critical examination before purchasing. 

The next thing to be considered is the use of the figures 
lines, and scales, as exhibited on the square. It is sup¬ 
posed that the ordinary divisions and sub-divisions of the 
inch, into halves, quarters, eighths, and sixteenths are un¬ 
derstood by the student; and that he also understands 
how to use that part of the square that is sub-divided into 
twelfths of an inch. This being conceded, we now proceed 
to describe the various rules as shown on all good squares; 
but before proceeding further, it may not be out of place 
to state, that on the tool recommended in this book, one 
sdge is subdivided into thirty-seconds of an inch. 

This fine sub-division will be found very useful, particu¬ 
larly so when used as a scale to measure drawings made in 


THE STEEL SQUARE 


18 

half, quarter, one-eighth, or one-sixteenth of an inch to the 
foot. 

I now refer the reader to the square shown in the 
Frontispiece. It is the one recommended in the foregoing 
pages, and is the most complete square in the market, and 
manufactured, I believe, but by one firm. It is known to 
the trade as No. ioo, and this number will be found 
stamped always on the face side of the square at the junc¬ 
tion of the tongue and blade. The following instructions 
refer to the Frontispiece and accompanying cuts. 

The diagonal scale is on the tongue at the junction with 
blade, Fig. i, and is for taking off hundredths of an inch. 
The lengths of the lines between the diagonal and the 



perpendicular are marked on the latter. Primary divisions 
are tenths, and the junction of the diagonal lines with the 
longitudinal parallel lines enables the operator to obtain 
divisions of one hundredth part of an inch; as, for example, 
if we wish to obtain twenty-four hundredths of an inch, we 
place the compasses on the “ dots ” on the fourth parallel 
line, which covers two primary divisions, and a fraction, or 















































AND ITS USES. 


I 9 


four-tenths of the third primary division, which added 
together makes twenty-four hundredths of an inch. 
Again, it we wish to obtain five tenths and seven hun¬ 
dredths, we operate on the seventh line, taking five 
primaries and the fraction of the sixth where the diagonal 
intersects the parallel line, as shown by the “dots,” on 
the compasses, and this gives us the distance required. 

Fig. 2 a shows the position of the “ dots ” or “ points ” 
referred to in the foregoing example. 

In practice it has been found that the diagonal scale 
is not so well adapted to tne use of the ordinary mechanic 
as the plain scale finely divided. On the squares now 
made the latter is used entirely as shown on Page 109. 

Board, Plank and Scantling Measure.— Perhaps, with 
the single exception of the common inch divisions on the 
square, no set of figures on the instrument will be found 
more useful to the active workman than that known as the 
board rule. A thorough knowledge of its use may be ob¬ 
tained by ten minutes’ study, and, when once obtained, is 
always at hand and ready for use. 

The following explanations are deemed sufficiently clear 
to give the reader a full knowledge of the workings of the 
rule. If we examine Fig. 2, in the Frontispiece, we will 
find under the figure 12, on the outer edge of the blade, 
where the length of the boards, plank, or scantling to be 
measured, is given, and the answer in feet and inches is found 
under the inches in width that the board, etc., measures. 
For example, take a board nine feet long and five 
inches wide; then under the figure 12, on the second 
line will be found the figure 9, which is the length 
of the board; then run along this line to the figure 


THE STEEL SQUARE 


to 

directly under the five inches (the width of the board), and 
we find three feet nine inches, which is the correct answer 
m “ board measure.” If the stuff is two inches thick, the 
sum is doubled; if three inches thick, it is trebled, etc., etc. 
If the stuff is longer than any figures shown on the square, 
\t can be measured by dividing and doubling the result. 
This rule is calculated, as its name indicates, for board 
measure, or for surfaces i inch in thickness. It may be 
advantageously used, however, upon timber by multiplying 
the result of the face measure of one side of a piece by its 
depth in inches. To illustrate, suppose it be required to 
measure a piece of timber 25 feet long, 10x14 inches in 
size. For the length we will take 12 and 13 feet. For 
the width we will take 10 inches, and multiply the result 
by 14. By the rule a board 12 feet long and io inches 



wide contains 10 feet, and one 13 feet long and 10 inches 
wide, 10 feet 10 inches. Therefore, a board 25 feet long 
and 10 inches wide must contain 20 feet and 10 inches. 
In the timber above described, however, we have what is 
equivalent to 14 such boards, and therefore we multiply 
this result by 14, which gives 291 feet and 8 inches, the 
board measure. 

The “ board measure,” as shown on the portion of the 









~*AND ITS USES. 


21 


square, Fig. 3, gives the feet contained in each board ac¬ 
cording to its length and width. This style of figuring 
squares, for board measure, is going out of date, as it gives 
the answer only in feet. 





1 ili iIlAriLij 


Fig. 3 a. 


Fig. 3 a shows the me-thod now in use for board measure. 
This shows the correct contents in feet and inches. It i$> 
a portion of the blade of the square, as shown at Fig. 2, or 
the Frontispiece. 


Brace Rule. —The “ brace rule ” is always placed on the 
tongue of the square, as shown in the central space at x, 
Fig. 1. 

This rule is easily understood; the figures on the left of 
the line represent the “ run ” or the length of two sides of a 
right angle, while the figures on the right represent the 
exact length of the third side of a right-angled triangle, in 
inches, tenths, and hundredths. Or, to explain it in another 
way, the equal numbers placed one above the other, may 
be considered as representing the sides of a square, and 




























22 


THE STEEL SQUARE 


the third number to the right the length of the diagonal of 
that square. Thus the exact length of a brace from point 
to point having a run of 33 inches on a post, and a run of 
the same on a girt, is 46-67 inches. The brace rule varies 
somewhat in the matter of the runs expressed in different 
squares. Some squares give a few brace lengths of which 
the runs unon the post and beam are unequal. 

Octagonal Scale. —The “ octagonal scale,” as shown on 
the central division of the upper portion of blade, is on the 
opposite side of the square to the “ brace rule,” and runs 
along the centre of the tongue as at s s. Its use is as fol¬ 
lows : Suppose a stick of timber ten inches square. Make 
a centre line, which will be five inches from each edge; set 
a pair of compasses, putting one leg on any of the main 
divisions shown on the square in this scale, and the other 
leg on the tenth subdivision. This division, pricked off 
from the centre line on the timber on each side, will give 
the points for the gauge-lines. Gauge from the corners 
both ways, and the lines for making the timber octagonal 
in its section are obtained. Always take the same number 
of spaces on your compasses as the timber is inches square 
from the centre line. Thus, if a stick is twelve inches 
square, take twelve spaces on the compasses; if only six 
inches square, take six spaces on the compasses, etc., etc* 
The rule always to be observed is as follows: Set off from 
each side of the centre line upon each face as many spaces 
by the octagon scale as the timber is inches square. For 
timbers larger in size than the number of divisions in the 
•cale, the measurements by it may be doubled or trebled, 
m the case may be. 


AND ITS USES. 


2 3 


The diagram, Fig. 4 a , shows the application of the rule 
applied to the end of a stick of timber or on a plane sur¬ 
face. Let b c d e, be the square equal to six inches > <n a 
side. Draw the centre lines, b c and D e, then with ti:« 



dividers take from the scale six parts, and lay off this dis¬ 
tance from the centre of each; as b 1, b 2, e 3 and E 4, 
c 5 and c 6, d 7 and d 8. Draw lines from 1 to 8, 2 to 3, 
4 to 5, 6 to 7, and the octagon figure is complete. 

A rule for laying off octagons is figured on nearly all 
carpenters’ two-foot rules, marked off from the inner edges 
of the rule; one set of figures is denoted by the letter e, 
another set is denoted by the letter m. That set marked 
e measures the distance from the edge of the square to the 
points indicated in the diagram, by the figures 1, 2, 3, 4, 
etc. The set marked m is used for finding the points 1, 2, 
3, 4, etc., by measuring from the middle or centre lines, b, 
E, C, D. 

have now fully described all the lines, figures, and 
scales that are usually found on the better class of squares 
now in use; but, I may as well here remark that there are 
squares in use of an inferior grade, that are somewhat dif- 









24 


THE STEEL SQUARE 


ferently figured. These tools, however, are such as can 
not be recommended for the purposes of the scientific 
carpenter or joiner. 

Fence. —A necessary appendage to the steel square in 
solving mechanical problems, is, what I call, for the want 
of a better name, an adjustable fence. This is made out 
of u piece of black walnut or cherry 2 inches wide, and 2 
feet 10 inches long (being cut so that it will pack in a tool 
chest), and 1^5 inches thick; run a gauge line down the 
centre of both edges; this done, run a saw kerf cutting 
down these gauge lines at least one foot from each end, 
leaving about ten inches of solid wood in the centre of 
fence. We now take our square and insert the blade in the 
saw kerf at one end of the fence, and the tongue in the 
kerf, at the other, the fence forming the third side of a * 
right-angle triangle, the blade and the tongue of the square 
forming, vhe other two sides. A fence may be made to do 

r— ; — ~ i 

Fig. 4. 


prett> fair service, if the saw kerf is all cut from one end 
as shown at Fig. 4. The one first described, however, will 
be found the most serviceable. The next step will be to 
make some provision for holding the fence tight on the 
square ; this is best done by putting a No. 10 i l / 2 inch 
screw in each end of the fence, close up to the blade and 
tongue; having done this, we are ready to proceed to busi¬ 


ness. 






AND ITS USES. 


2 5 


Application. —The fence being made as desired, in either 
of the methods mentioned, and adjusted to the square, work 
can be commenced forthwith. 

The first attempt will be to make a pattern for a brace, 
for a four-foot “run.” Take a piece of stuff already pre¬ 
pared, six feet long, four inches wide and half-inch thick, 
gauge it three-eighths from jointed edge. 

Take the square as arranged at Fig. 5, and place it on 
the prepared stuff as shown at Fig. 6. Adjust the square 
so that the twelve-inch lines coincide exactly with the 
gauge line o, o, o, c Hold the square firmly in the posi¬ 
tion now obtained, and slide the fence up the tongue and 
blade until it fits snugly against the jointed edge of the 
prepared stuff, screw the fence tight on the square, and be 
sure that the 12 inch marks on both the blade and the 
tongue are in exact position over the gauge-line. 

I repeat this caution, because the successful completion 
of the work depends on exactness at this stage. 

We are now ready to lay out ihe pattern. Slide the 
square to the extreme leffe, as shown on the dotted lines at 
x, mark with a knife on the outside edges of the square, 
cutting the gauge-line. Slide the square to the right until 
the 12 inch mark on the tongue stands over the knife mark 
on the gauge-line; mark the right-hand side of the square 
cutting the gauge-line as before, repeat the process four 
times, marking the extreme ends to cut off, and we have 
the length of the brace and the bevels. 

Square over, with a try square, at each end from the 
gauge-line, and we have the toe of the brace. The lines, 
s, s, shown at the ends of the pattern, represent the tenons 
that are to be left on the braces. This pattern is now com- 


26 


THE STEEL SQUARE 


#• 


































































AND ITS USES. 


2 7 


plete; to make it handy for use, however, nail a strip 2 inches 
wide on its edge, to answer for a fence as shown at K., and 
the pattern can then be used either side up. 

The cut at Fig. 7, shows the brace in position, on a re¬ 
duced scale. The principle on which the square works in 
the formation of a brace can easily be understood from this 
cut, as the dotted lines show the position the square was in 
when the pattern was laid out. 

It may be necessary to state that the “ square,” as now 
arranged, will lay out a brace pattern for any length, if the 
angle is right, and the run equal. Should the brace be of 
great length, however, additional care must be taken in the 
adjustment of the square, for should there be any departure 
from truth, that departure will be repeated every time the 
square is moved, and where it would not affect a short run, it 
might seriously affect a long one. 

To lay out a pattern for a brace where the run on the 
beam is three feet, and the run down the post four, proceed 
as follows: 

Prepare a piece of stuff, same as the one operated on for 
four feet run; joint and gauge it. Lay the square on the 
left-hand side, keep the 12 inch mark on the tongue, over 
the gauge-line, place the 9 inch mark on the blade, on the 
gauge-line, so that the gauge-line forms the third side of a 
right-angle triangle, the other sides of which are nine and 
twelve inches respectively. 

Now proceed as on the former occasion, and as shown 
at Fig. 8, taking care to mark the bevels at the extreme 
ends. The dotted lines show the positions of the square, 
as the pattern is being laid out. 

Fig. q shows the brace in position, the dotted lines show 


28 


‘the steel square 


where the square was placed on the pattern. It is well to 
thoroughly understand the method of obtaining the lengths 
and bevels of irregular braces. A little study, will soon 
enable any person to make all kinds of braces. 

If we want a brace with 
a two feet run, and a four 
feet run, it must be evident 
that, as two is the half of 
four, so on the square take 
12 inches on the tongue, and 
6 inches on the blade, apply 
four times, and we have the 
length, and the bevels of a 
brace for this run. 

For a three by four feet 
run, take 12 inches on the 
tongue, and 9 inches on the 
blade, and apply four times, 
because, as 3 feet is ^ of four 
feet, so 9 inches is of 12 
inches. 

Rafters. —Fig. 10 shows 
a plan of a roof, having 
twenty-six feet of a span. 

The span of a roof is the 
distance over the wall plates 
measuring from A to A, as shown in Fig. 10. It is also 
the extent of an arch between its abutments. 

Ihere are two rafters shown in position on Fig. 10. The 
one on the left is at an inclination of quarter pitch, and 



o 






AND ITS USES. 

marked B, and the one on the right, 
marked C, has an inclination of one-third 
pitch. These angles, or inclinations rather, 
are called quarter and third pitch, respec¬ 
tively, because the height from level of wall 
plates to ridge of roof is one-quarter or one- 
third the width of building, as the case 
may be. 

At Fig. n, the rafter B is shown drawn to 
a larger scale; you will notice that this rafter 
is for quarter pitch, and for convenience, it is 
supposed to consist of a piece of stuff 2 
inches by 6 inches by 17 feet. That portion 
of the rafter that projects over the wall of the 
building, and forms the eve, is three or more 3 

O 

inches in width, just as we please. The * M 

M 

length of the projecting piece in this case is 
one foot—it may be more or less to suit the 
eve, but the line must continue from end to 
end of the rafter, as shown on the plan, and 
we will call this line our working line. 

We are now ready to lay out this 
rafter, and will proceed as follows: We 
adjust the fence on the square the same as 
for braces, press the fence firmly against 
the top edge of rafter, and place the figure 
12 inches on the left-hand side, and the 
figure 6 in on the right-hand side, directly 
over the working line, as shown on the 
plan. Be very exact about getting the 
figures on the line, for the quality of the 














THE STEEL SQUARE 


3° 


work depends much on this; when you are satisfied thac 
you are right, screw your fence tight to the square. Com¬ 
mence at No. i on the left, and mark off on the working 
line; then slide your square to No. 2, repeat the marking 
and continue the process until you have measured off 
thirteen spaces, the same as shown by the dotted lines in 
the drawing. The last line on the right-hand side will be 
the plumb cut of the rafter, and the exact length required. 
It will be noticed that the square has been applied to the 
timber thirteen times. 

The reason for this is, that the building is twenty-six feet 
wide, the half of which is thirteen feet, the distance that 
one rafter is expected to reach, so, if the building was thirty 
feet w^de, we should be obliged to apply the square fifteen 
times instead of thirteen. We may take it for granted, 
then, that in all cases where this method is employed to 
obtain the lengths and bevels, or cuts of rafters, we must 
apply the square half as many times as there are feet in the 
w’.dth of the building being covered. If the roof to be 
covered is one-third pitch, all to be done is to take 12 
inches on one side of the square and 8 inches on the other, 
and operate as for quarter pitch. 

We shall frequently meet with roofs much more acute 
.han the ones shown, but it will be easy to see how they 
can be managed. For instance, where the rafters are at 
right-angles to each other, apply the square the same ax 
for braces of equal run, that is to say, keep the 12 mark on 
the blade, and the 12 mark on the tongue, on the working 
line. When a roof is more acute, or “steeper” than a 
right-angle, take a greater figure than twelve on one skU 
of the square, and twelve on the other. 


AND ITS USES. 


31 


Whenever a drawing of a roof is to be followed, we can 
soon find out how to employ the square, by laying it on 
the drawing, as shown in Fig. 12. Of course, something 
depends on the scale to which the drawing is made. II 
any of the ordinary fractions 
of an inch are used, the intelli¬ 
gent workman will have no 
difficulty in discovering what 
figures to make use of to get 
the “ cuts ” and length de¬ 
sired. 

Sometimes there may be a 
fraction of a foot in this divis¬ 
ion ; when such is the case, it 
can be dealt with as follows: 
suppose there is a fraction of 
a foot, say 8 inches, the half 
of which would be 4 inches, 
or yi of a foot; then, if the 
roof is quarter pitch, all to be 
done is to place the square, 
with the 4 inch mark on the 
blade, and the 2 inch mark on 
the tongue, on the centre line 
of the rafter, and the distance 
between these points is the 
extra length required, and the line down the tongue is the 
bevel at the point of the rafter. On Fig. 13, is snown 
an application of this method. All other pitches and frac¬ 
tions can be treated in this manner without overtaxing the 
ingenuity of the workman. 









3 * 


THE STEEL SQUARE 



F ig. 13. 


VERTICAL BOARD 




































AND ITS USES. 


33 


Sufficient has been shown to enable the student:, if he 
has mastered it, to find the lengths and bevels of any com¬ 
mon rafter; therefore, for the present, we will leave saddle 
roofs, and try what can be done with the square in de¬ 
termining the lengths and bevels of “ hips,” valleys, and 
cripples. 



Fig. 14 shows how to get bevels on the top end of vertical 
boarding, at the gable ends, suitable for the quarter pitch 
at Fig. 10. 

At Fig. 15, is shown a method for finding the bevel for 
horizontal boarding, collar ties, etc. 

Hip Rafters. —Fig. 16, is supposed to be the pitch of a 
roof furnished by an architect, with the square applied to 
the pitch. The end of the long blade must only just enter 






















34 


THE STEEL SQUARE 




















































AND ITS USES. 


35 


the fence, as shown in the drawing, and the short end must 
be adjusted to the pitch of the roof, whatever it may be. 
Fig. 17 shows the square set to the pitch of the hip rafter. 
The squares as set give the plumb and level cuts. Fig. 18 
is the rafter plan of a house 18 by 24 feet; the rafters are^ 
laid off on the level, and measure nine feet from centre of J 
ridge to outside of wall; there should be a rafter pattern ■ 
with a plumb cut at one end, and the foot cut at the other, 
got out as previously shown. (Figs. 16,17, 18, P.) When the 
rafter foot is marked, place the end of the long blade of the 
square to the wall line, as in drawing, and mark across the 
rafter at the outside of the short blade, and these marks on 
the rafter pitch will correspond with two feet on the level 
plan, slide the square up the rafter and place the end of the 
long blade to the mark last made, and mark outside the short 
blade as before, repeat the application until nine feet are 
measured off, and then the length of the rafter is correct; 
remember to mark off one-half the thickness of ridge-piece. 
The rafters are laid off on part of plan to show the appearance 
of the rafters in a roof of this kind, but for working purposes 
the rafters 1, 2, 3, 4, 5, and 6, with one hip rafter, is all 
that is required. 

Hip-roof Framing. —We first lay off common rafter, 
which has been previously explained; but deeming it ne¬ 
cessary to give a formula in figures to avoid making a plan, 
we take pitch. This pitch is *4 the width of building, to 
point of rafter from wall plate or base. For an example, 
always use 8, which is of 24, on tongues for altitude; 12, 
y 2 the width of 24, on blade for base. This cuts common 
r after. Next is the hip-rafter. It must be understood that 


Zr* 


THE STEEL SQUARE 


the diagonal of 12 and 12 is 17 in framing, and the hip is 
the diagonal of a square added to the rise of roof; there¬ 
fore we take 8 on tongue and 17 on blade; run the same 
number of times as common rafter (rule to find distance 
of hip diagonal a 2 + a 2 -f b 2 = y 2 ). To cut jack rafters, divide 
the numbers of openings for common rafter. Suppose we 
have 5 jacks, with six openings, our common rafter 12 feet 
long, each jack would be 2 feet shorter. First 10 feet, 
second 8 feet, third 6 feet, and so on. The top down cut 
the same as cut of common rafter; foot also the same. 
To cut mitre to fit hip. Take half the width of building 
on tongue and length of common rafter on blade; blade 
gives cut. Now find the diagonal of 8 and 12, which 
is 14 42 , call it 14 7-16, take 12 on tongue, 14 7-16 on 
blade; blade gives cut. The hip-rafter must be beveled to 
suit jacks; height of hip on tongue, length of hip on blade; 
tongue gives bevel. Then we take 8 on tongue 18^ on 
blade; tongue gives the bevel. Those figures will span all 
cuts in putting on cornice and sheathing. To cut bed 
moulds for gable to fit under cornice, take half width of 
building on tongue length o<" common rafter on blade; 
blade gives cut; machine mouldings will not member, but 
this gives a solid joint; and to member properly it is neces¬ 
sary to make moulding by hand, the diagonal plumb cut 
differences. I find a great many mechanics puzzled to 
makes the cuts for a valley. To cut planceer, to run up 
valley, take heighth of rafter on tongue, length of rafter on 
blade; tongue gives cut. The plumb cut takes the height 
of hip-rafter on tongue, length of hip-rafter on blade; 
tongue gives cut. These figures give the cuts for ]/i pitch 
only, regardless of width of building. - 


AND ITS USES. 


For a hopper the mitre is cut on the same principle. 
To make a butt joint, take the width of side on blade, and 
half the flare on tongue; the latter gives the cut. You 
will observe that a hip-roof is the same as a hopper in¬ 
verted. The cuts for the edges of the pieces of a hexagonal 
hopper are found this way: Subtract the width of one 
piece at the bottom from the width of same at top, take 
remainder on tongue, depth of side on blaclh; tongue gives 
the cut. The cut on the face of sides: Take 7-12 of the 
rise on tongues and the depth of side on blade; tongue 
gives cut. The bevel of top and bottom: Take rise on 
blade, run on tongue; tongue gives cut. 

Fig. 19 exhibits two methods of finding the “ backing ” 
of the angle on hip-rafter. The methods are as simple as 
any known. Take the length of the rafter on the blade, 
and the rise on the short blade or tongue, place the 
square on the line D E, the plan of the hip, the angle is 
given to bevel the hip-rafter, as shown at F. This method 
gives the angle, only for a right-angled plan, where the 
pitches are the same, and no other . 

The other method applies to right, obtuse, and acute 
angles, where the pitches are the same. At the angle d 
will be seen the line from the points k l, at the intersec¬ 
tion of the sides of the angle rafter with the sides of the 
plan. 

With one point of the compass at d, describe the curve 
from the line as shown. Tangential to the curve draw 
the dotted line, cutting a, then draw a line parallel to a b, 
the pitch of the hip. The pitcn or bevel, will be found 
at G, which is a section of the hip-rafter. 

This problem is taken from “ Gould's Carpenters’ 


3« 


THE STEEL SQUARE 







Fig. iq. 














AND ITS USES. 39 

Guide,” but has been in practice among workmen for 
many years. 



Fig. 20 exhibits a method of finding the cuts in a mitre 
box, by placing the square on the line a b at equal dis¬ 
tances from the heel of the square, say ten inches. The 
bevel is shown to prove the truth of the lines by applying 
it to opposite sides of the square. 


Stairs. —In laying out stairs with the square, it is neces¬ 
sary to first determine the height from the top of the floor 
on which the stairs start from, to the floor on which they 
are to land ; also the “ run ” or the distance of their hori¬ 
zontal stretch. These lengths being obtained, the rest is 
easy. 

Fig. 21 shows a part of a stair string, with the “ square ” 
laid on, showing its application in cutting out a pitch-board. 
As the square is placed it shows io inches for the tread and 
7 inches for the rise. 

To cut a pitch-board, after the tread and rise have beeD 
























40 


THE STEEL SQUARE 


determined, proceed as follows: Take a piece of thin, clear 
stuff, and lav the sauare on the face edge, as shown in the 
figure, and mark out the pitch-board with a sharp knife \ 



then cut out with a fine saw and dress to knife marks, nail 
a piece on the longest edge of the pitch-board for a fence, 
and it is ready for use. 

Fig. 22 is a rod, with the dumber and heighth of steps 
for a rough flight of stairs to lead down into a cellar or 
elsewhere. 

Fig. 23 is a step-ladder, sufficiently inclined to permit a 
person to pass up and down on it with convenience. To 
lay off the treads, level across the pitch of the ladder, set 
the short side ot rhe square on the floor, at the foot of the 
string, after the string is cut, to fit the floor and trimmer 
joists. Fasten the fence on the square, as shown at Fig. 5. 
The height of the steps in this case is nine inches, so it will 
be seen that it is an easy matter to lay off the string, as the 























































































AND ITS USES, 


TOP OF JOIST 













































































42 


THE STEEL SQUARE 


long side of the square hangs plumb, and nine inches up 
its length will be the distance from one step to the next 
one. 

Fig. 24 shows the square and fence in position on the 
string. 

The opening in the floor at the top of the string shows 
the ends of trimming joists, five feet apart. • 

Fig. 25 shows how to divide a board into an even number 
of parts, each part being equal, when the same is an un¬ 
even number of inches, or parts of an inch in width. Lay 
the square as shown, with the ends of the square on the 
edges of the board, then the points of division will be 
found at 6, 12, and 18, for dividing the board in foui 
equal parts; or at 4, 8, 12, 16, and 20, if it is desired to 
divide the board into six equal parts. Of course, the 
common two-foot rule will answer this purpose as well 
as the square, but it is not always convenient. 

Fig. 26 shows how a circle can be described by means 
of a “ steel square ” without having recourse to its centre. 

At the extremities of the diameter, a, o, fix two pins, as 
shown; then by sliding the sides of the square in contact 
with the pins, and holding a pencil at the point x, a semi¬ 
circle will be struck. Reverse the square, repeat the pro¬ 
cess, and the circle is complete. 

Miscellaneous Rules —The following rules have been 
tested over and over again by the writer, and found reliable 
in every instance. They have been known to advanced 
workmen for many years, but were never published, so far 
as the writer knows, until they appeared in the Builder and 
Wood- Worker , some years ago: 


AND ITS USES. 


43 


Measurement. —Let us suppose that we have a pile of 
lumber to measure, the boards being of different widths, and 
say 16 feet long. We take our square and a bevel with a 
long blade and proceed as follows : First we set the bevel 
at 12 inches on the tongue of the square, because we want 
to find the contents of the board in feet, 12 inches being 
one foot; now we set the other end of the bevel blade on the 
16 inch mark on the blade of the square, because the boards 
are 16 feet long. Now, it must be quite evident to any 
one who would think for a moment, that a board 12 inches, 
or one foot wide, and 16 feet long, must contain 16 feet of 
lumber. Very w r ell, then we have 16, the length, on the 
blade. Now, we have a board n inches wide, we just 
move our bevel from the 12 inch mark to the 11 inch mark, 
and look on the blade of the square for the true answer; 
and so on with any width, so long as the stuff is 16 feet 
long. If the stuff is 2 inches thick, double the answer, if 
3 inches thick, treble the answer, etc. 

Now, if we have stuff 14 feet long, we simply change 
the bevel blade from 16 inches on the square blade, to 14 
inches, keeping the other end of the bevel on the 12 inch 
mark, 12 inches being the constant figure on that side of 
the square, and it will easily be seen that any length of 
stuff within the range of the square can be measured ac¬ 
curately by this method. 

If we want to find out how many yards of plastering or 
painting there are in a wall, it can be done by this method 
quite easily. Let us suppose a wall to be 12 feet high and 
18 feet long, and we want to find out how many yards of 
{Mastering or painting there are in it, we set the bevd on 
the 9 inch mark on the tongue (we take 9 inches because 9 


44 


THE STEEL SQUARE 


square feet make one square yard,) we take 18 inches, one 
of the dimensions of the wall, on the blade of the square; 
then after screwing the bevel tight, we slide it from 9 inches 
to 12 inches, the latter number being the other dimension, 
and the answer will be found on the blade of the square. 
It must be understood that 9 inches must be a constant 
figure when you want the answer to be in yards, and in 
measuring for plastering it is as well to set the other end of 
the bevel on the figure that corresponds with the height of 
the ceiling, and then there will require no movement of 
the bevel further than to place it on the third dimension. 
This last rule is a very simple and very useful one; of course 
“ openings ” will have to be allowed for, as this rule gives 
the whole measurement. 

If the diagonal of any parallelogram within the range of 
the square is required, it can be obtained as follows: Set the 
blade of the bevel on 8^ in. on the tongue of the square, 
and at 12^4 in. on the blade; securely fasten the bevel at this 
angle. Now, suppose the parallelogram or square to be n 
inches on the side, then move the bevel to the 11 inch mark 
on the tongue of the square, and the answer, 15 9-16, will be 
found on the blade. All problems of this nature can be solved 
with the square and bevel as the latt-er is now set. There 
is no particular reason for using 8^ and 12^4, only that 
they are in exact proportion to 70 and 99. 4^4 and 6 

3-16 would do just as well, but would not admit as ready 
an adjustment of the bevel. 

To find the circumference of a circle with the square and 
bevel proceed as follows: Set the bevel to 7 on the tongue 
and 22 on the blade; move the bevel to the given diameter 
on the tongue of the square, and the approximate answer 


AND ITS USES. 


45 


will be found on the blade. When the circumference is 
wanted the operation is simply reversed, that is, we put 
the bevel on the blade and look on the tongue of the square 
for the answer. 

If we want to find the side of the greatest square that 
can be inscribed in a given circle, when the diameter is 
given, we set the bevel to 8)4 on the tongue and 12 on the 
blade. Then set the bevel of the diameter, on the blade, 
and the answer will be found on the tongue. 

The circumference of an ellipse or oval is found by set¬ 
ting 5^3 inches on the tongue and 8)4 inches on the blade; 
then set the bevel to the sum of the longest and shortest 
diameters on the tongue, and the blade gives the answer. 

To find a square of equal area to a given circle, we set the 
bevel to 9^ inches on the tongue, and 11 inches on the blade; 
then move the bevel to the diameter of the circle on the 
blade, and the answer will be found on the tongue. If the 
circumference of the circle is given, and we want to find a 
square containing the same area, we set the bevel to 5^ 
inches on the tongue and 19^4 inches on the blade. 

On Fig. 27 is shown a method to determine the pro¬ 
portions of any circular presses or other cylindrical bodies, 
by the use of the square. Suppose the small circle, n, to 
be five inches in diameter and the circle r is ten inches in 
diameter, and it is required to make another circle, z, to 
contain the same area as the two circles n and r. Meas¬ 
ure line a, on the square d, from five on the tongue to 10 
on the blade, and the length of this line a from the two 
points named will be the diameter of the larger circle z. 
And again, if you want to run these circles into a fourth 
one, set the diameter of the third on the tongue of the square, 


46 


THE STEEL SQUARE 


and the diameter of z on the blade, and the diagonal 
will give the diameter of the fourth or largest circle, and the 
same rule may be carried out to infinite extent. The rule 
is reversed by taking the diameter of the greater circle and 
laying diagonally on the square, and letting the ends touch 



whatever points on the outside edge of the square. These 
points will give the diameter of two circles, which com¬ 
bined, will contain the same area as the larger circle. The 
same rule can also be applied to squares, cubes, triangles, 
rectangles, and all other regular figures, by taking similar 
dimensions only; that is, if the largest side of one triangle 
is taken, the largest side of the other must also be taken, 
and the result will be the largest side of the required tri¬ 
angle, and so with the shortest side. 

In Fig. 28 we show how the centre of a circle may be de¬ 
termined without the use of compasses; this is based on 
the principle that a circle can be drawn through any three 
points that are not actually in a straight line. Suppose we 
take a b c D for four given points, then draw a line from a 









AND ITS USEK. 


47 


to d, and from b to c; get the centre of these lines, and 
square from these centres as shown, and when the square 
crosses, the line, or where the lines intersect, as at x, there 
will be the centre of the circle. This is a very useful rule, and 



Fig. 28. 


by keeping it in mind the mechanic may very frequently 
save himself much trouble, as it often happens that it is ne¬ 
cessary to find the centre of the circle, when the compasses 
are not at hand. 

In Fig. 29 we show how the square can be used, in lieu 
of the trammel, for the production of ellipses. Here the 
square, e d f, is used to form the elliptical quadrant, 
a b, instead of the cross of the trammel; h l k may be 
simply pins, which can be pressed against the sides of the 
square while the tracer is moved. In this case the adjust¬ 
ment is obtained by making the distance, h /, equal to the 
semi-axis minor, and the distance l k , equal to the semi-axis 
major. 




4 8 


THE STEEL SQUARE 



Fig. 30 shows a method of describing a parabola bv 
means of a straight rule and a square, its double ordinate 
and abscissa being given. Let a c be the double ordinate, 
andD b the abscissa. Bisect d c in f; join b f, and draw 
f e perpendicular to b f, cutting the axis b d produced in , 
f. From b set off b g equal to d e, and g will be the focus 
of the parabola. Make b l equal to b g, and lay the rule 
on straight-edge h k on l, and parallel to a c. Take a 
string, mfg, equal in length toL e; attach one of its ends 
to a pin, or other fastening, at g, and its other end to the 
end m, of the square m n o. If now the square be slid 
along the straight-edge, and the string be pressed against 










AND ITS USES. 


49 

its edge m n, a pencil placed in the bight at f will describe 
the curve. 



The two arms of a horizontal lever are respectively 
9 inches and 13 inches in length from the suspending 
point; a weight of 10 lbs. is suspended from the shorter 
arm, and it is required to know what weight will be re¬ 
quired to suspend on the long arm to make it balance. 
Set a bevel on the blade of square at 13 inches and the 
other end of the bevel on the 9 inch mark on tongue of 
square, then slide the bevel from 13 inches to 10 on the 
blade of square, and the answer will be found on the 
tongue of the square. It is easy to see how this rule can 
be reversed so that a weight required for the shorter arm 
can be found. 

Fig. 31 shows how to get the flare for a hopper 4 feet 
across the top and 16 inches perpendicular depth. Add to 
the depth one-third of the required size of the discharge 











s° 


THE STEEL SQUARE 



Fig. 31. 


hole (the draft represents a 6-inch hole), which makes 18 
inches, which is represented on the tongue of the square. 
(The figures on the draft are 9 and 12, which produce the 
same bevel.) Then take one-half, 24 inches of the width 
across the top of the hopper, which is represented on the 
blade of the square. Then scribe along the blade as rep¬ 
resented by the dotted lines, which gives the required flare. 
(The one-third added to the depth is near enough for 
all practical purpose for the discharge.) 



Fig. w. 













































AND ITS USES. 


5 1 

Fig. 32 shows how to apply the square to the edge of 
a board in order to obtain the bevel to form the joint. 
Using the same figures as in Fig. 31, scribe across the edge 
of the board by the side of the tongue, as shown by dotted 
lines. The long point being the outside. 



Fig. 33. 


On Fig. 33 we show a quick method of finding the 
centre of a circle: Let n n, the corner of the square, touch 
the circumference, and where the blade and tongue cross 
it will be divided equally; then move the square to any 
other place and mark in the same way and straight edge 
across, and where the line crosses a, b, as at o, there will 
be the centre of the circle. 










5 2 


THE STEEL SQUARE 


i and 2, Fig. 34, are taken from Gould’s Wood- Work¬ 
ing Guide. 

The portion marked a, exhibits a method of finding the 
lines for eight-squaring a piece of timber with the square, 

by placing the block on the 
piece, and making the points 
seven inches from the ends 
of, the square, from which to 
draw the lines for the sides 
of the octagonal piece re¬ 
quired. 

At the heel of the square 
is shown a method of cut¬ 
ting a board to fit any angle 
with the square and compass, 
by placing the square in the 
angle, and taking the distance 
from the heel of the square 
to the angle a, in the com- 
)ass; then lay the square on 
the piece to be fitted, with 
the distance taken, and from 
the point a, draw the line a 
b, which will give the angle 
to cut the piece required. 

At 2 is shown a method 
of constructing a polygonal 
figure of eight sides; by placing the square on the line a b, 
with equal distances on the blade and tongue, as shown; 
the curve lines show the method of transferring the dis¬ 
tances; the diagonal gives the intersection at the angles. 



O 
i *1 



























































AND ITS USES. 


iS 


There are at least a dozen different ways of forming oc¬ 
tagonal figures by the square; some of them are tedious 
and difficult, while others can not be applied under all cir¬ 
cumstances. The method shown at Fig. 35 is handy and 
easily understood. 



F*g- 35 - 


An equilateral triangle can be formed by taking half of 
one side on the tongue of the square, as shown at Fig. 36. 
The line along the edge of the tongue forms the mitre for 



Fig. 36. 

the triangle, and the line along the edge of the blade forms 
e cut for the jomts of a hexagon, and as six equi- 












34 


THE STEEL SQUARE 


lateral triangles form a hexagon when one point of each is 
placed at a central point, o, it follows that a hexagon may 
be constructed by the square above. 

The following is a good method for obtaining the cuts 
for a horizontal and raking cornice; it is correct and simple ; 
the gutter to be always cut a square mitre. 

The seat or run of the rafter on the blade, r c, Fig. 37, 
the rise of the roof on the tongue, a c, mark against the 
tongue, gives the cut for the side of the box, a c. The 



diagonal a, r, which is the length of the rafter on the blade 
a, d, the seat of the rafter on the tongue d, s, mark against 
the blade gives the cut across the box, ad. d a c is the 
mitre cut to fit the gutter; then if we square across the 
box from a, it gives f, a, c the cut for the gable peak. 

At Fig. 38 is shown a method for obtaining either the 
butt or mitre cuts, for “ Hopper” work. 

The line, s s, in the cut represents the edge of a board; 
the line, a b, the flare of hopper. Lay the square on the 
face of the board so that the blade will coincide with flare 
of hopper, a b, then mark by the tongue the line b c, then 
square from edge of board, s s, cutting the angle b. 

Now we have a figure that will, when used on the steel 


































AND ITS USES. 


5b 


square, give the cuts for a hopper of any flare, either with 
butt or mitre joints. 

To find bevel to cut across face of board: 

Take a b on blade and a d on tongue, bevel of tongue 
is the bevel required. 



To find the bevel for butt-joint: Take b c on blade 
*^d A d on tongue; bevel of tongue is the bevel required. 

To find the bevel for mitre joint: Take b c on blade 
and D C on tongue; bevel of tongue is the bevel required. 

It will be seen that this is a very simple method of 
solving what is usually considered a very difficult problem. 





S6 


I 


THE STEEL SQUARE 


PART II. 

The following useful applications of the square were 
kindly furnished for this work, by Mr. Croker; several of 
them are new and original: 

Consider the blade of the square as representing the 
span of a building, but without any reference to actual or 
scale measurement. Next, some particular portion of the 
blade is to be taken as the rise of the supposed building; if 
a third, fourth, or half pitch is required, then a third, 
fourth, or a half of the blade is conceived as the rise which 
with half the blade solves the pitch. From this it will be 
seen that half the blade is always taken as the base of the 
theoretical common rafter. Where we have to deal with 
irregular pitches—by which is meant those pitches which 
are not a quarter, sixth, third, half, etc., of the building— 
then the square is to be applied to the irregular pitch 
with the blade lying in the direction of the pitch and 
the centre of the blade at the intersection of pitch and 
base line of the common rafter, and the resulting distance 
on the tongue, where it intersects the base line, is the 
distance to be taken as the rise of the theoretical rafter. 
Let us now take a hip-roof over a square plan (for all the 
rules apply only to square planned building), and the prac¬ 
tical problems supposed to need solution are: Length of 
common rafters, the plumb and level cuts; length of hip- 
rafter, its plumb and level cuts; bevel of jacks and sheet- 





AND ITS USES. 


57 


ing boards against the hips; “ backing ” of the hip-rafter, 
top and down bevel of a purlin mitering against the hip 
with its surface in line with the plane of the roof. If the 
student can readily and intelligently solve these problems, 
he will be in a position to make extensions in the principles 
involved. Let the width of building under consideration 
be 24 feet wide, and of one third pitch. 



Let 1, 12. Fig. 36, be the base of the theoretical common 
rafter, eight inches rise, equal to one third of the blade, be¬ 
cause it is a third pitch; mark along the blade and extend 
the heel, making it and 12 equal to half the width of 
the actual building to a scale of 1^ inch to a foot; this is a 
much better scale to work by than an inch one, being larger 
and more legible, eighths being inches, sixteenths, y 2 inches, 
etc., thus enabling very accurate measurements to be taken. 
By the way, it is a good plan to have the square stamped 
off on the eighths side at every 1 y 2 inches for feet, for more 
readily counting the scale; then mark along the tongue at 
b, which gives b 12 the length of common rafter; level cut 
on blade and plumb cut on tongue. Next take the rise of 
the theoretical common rafter on the tongue, and 17 inches 





5« 


THE STEEL SQUARE 




on the blade, as the theoretical base of the hip-rafter; 
place the square as shown at Fig. 37 ; then multiply the 



actual base of common rafter 12, (Fig. 36.) by 1*414=5 
16*968 feet, or 17 feet, practically, which set off on blade at A 
17 ; mark on tongue at b, then b i 7 is the length of hip- 
rafter. For the bevels of jacks and sheeting-boards against 
hips take the diagonal b 12 —theoretical rafter—Fig. 36, on 
the blade with half the blade—the theoretical base— and 
place the square as shown at Fig. 38, then mark along the 
blade for bevel of jacks, and along tongue for bevel of 
sheeting-boards. 



Fig. ji. 












AND ITS USES. 


59 


For the “ backing ” of hip, take the diagonal of the 
theoretical hip-rafter, 8, 17 (Fig 37), on the blade, and its 
rise—8 inches—on the tongue, and place square as shown 
ft* Fig. 39; mark by the tongue which gives the bevel re¬ 



quired. To get the upper bevel of a purlin lying in 
the plane of the roof, take the bevel at tongue (Fig. 38), 
for the down bevel take the blade distance 147—16 (Fig. 
38) on the blade with the theoretical rise—8; place the 
square as shown at Fig. 40; mark by the tongue which 
gives the bevel required. 



Fig. 41 shows how any length or breadth within the extent 
of the blade of the square can be instantly divided into any 
equal parts. Let a and b represent the edges of a board, say 
8^ inches, wide, to be divided into 5 equal parts; take any 






THE STEEL SQUARE 


convenient 5 parts, say 15 inches, because 5X3 = 15, placing 
heel of square fair to edge b, and 15 to edge a ; mark off at 
every 3 inches on blade, as shown, and draw lines through 
these points, which will divide the board as required. We 
will here show how the square can be used to solve problems 
in proportion; for instance, if 1500 feet of boards cost 
$10.75, what will 600 feet cost? Take 15 on the blade 


B 



and 10*75 on the tongue, and place the square as shown at 
Fig. 41, then count from 15 towards b, and from this point 
draw parallel to tongue; 6 a, this is the answer re- 
quired. 



Fig. 42. 


Fig. 43. 




















AND 1 IS USES. 


61 


yigs. 42 and 43 show quite a novel and useful way of 
bisecting any angle. Let a 12, a b be the given sides of an 
acute angle to be bisected. At any convenient point as c 
square c d from c 12. Now take c d on the tongue, and 
the sum of a d and a c on the blade of the square, place 
as shown in the Figure, then mark by the blade, which is 
the bisection required. If the angle is obtuse, as a b, a f, 
(Fig. 42), produce a convenient distance, as a c, square 
over c d, take c D on the tongue, and the sum of a d, a c, 
on the blade, place square as shown, and mark by the 
tongue for the required bisection. 



Fig. 44 shows a handy way of finding the bevel of rails 
to diminish door stiles. Place the square fair with the 
known joint a b, mark by the tongue, then the resulting 
be^ei at a ** the same as that at B. 













62 


THE STEEL SQUARE 


PART III. 

* 

The following rules have been gathered from vtuious 
sources, chiefly, however, from papers recently published in 
the Scientific American Supplement, by John O. Connell, of 
St. Louis, and from papers contributed to the Builder and 
Wood-Worker, by Wm. E. Hill, of Terre Haute, Ind. 



* Fig. 45 shows how an octagon can be produced by the 
aid of a steel square. Prick off the distance a o equal to 
half the distance of the square; mark this distance on the 
blade of the square from b to o, place the square on the 


* Wm. E. Hill 





Fig. 46, 


AND ITS USES. 


63 





diagonal, as shown, and 
square over each way. Do 
the same at every angle, 
and the octagon is com¬ 
plete. 

To obtain the same figure 
with the compasses*, pro¬ 
ceed as follows : Take half 
the diagonal on the com¬ 
passes, make a little over a 
quarter sweep from c, and 
at the insersection at d and 
c, then d and c form one 
side of an octagonal figure. 

Again: take a piece of 
timber twelve inches square, 
as at Fig. 46 ; take twelve 
inches on the blade and 
tongue from a to b, and a 
to c, mark at the point a, 
operate similarly on the op¬ 
posite edge, and the marked 
points will be guides for 
guage-lines for the angles 
forming an octagon. The 
remaining three sides of the 
timber can be treated in 
the same manner. 

These points can be 
found with a carpenter’s 
rule as follows: Lay tb* 












64 the steel square 

rule on the timber, partly opened, as shown, in the cut, 
“prick off” at the figures 7 and 17 as at a and b, and 
these points will be the guides for the gauge-lines. The 
same points can be found by laying the square diagonally 
across the timber and “ pricking ” off 7 and 17. 

To make a moulder’s flask octagonal proceed as follows : 
The flask to be four feet across. Multiply 4 X 5 (as an 
octagon is always as 5 to 12 nearly), which gives 20; di¬ 
vide by 12, which gives 1^ feet, cut mitre to suit this 
measurement, nail into corners of square box, and you have 
an octagon flask at once. 

Another method of constructing an octagon is shown at 
Fig. 47. Take the side as a b for a radius, describe an arc 



cutting the diagonal at d ; square over from d to and 
the point e will then be the gauge-guide for all the sides. 











AND ITS USES. 




Another method (Fig. 48) is to draw a straight line, c 
A any length; then let a b and a c be corresponding 
figures on the blade and tongue of the square, mark along 
either and measure the distance of required octagon; move 



the square and mark also. Now use the square the 
same as before, and the marks c b and b d are the points 
required. 

Fig. 49 shows the application of a long bevel to a 
square, by which some calculations can be made with 
greater ease and quickness than by the usual arithmetical 
process. The largest size of carpenter’s bevel placed under 
the framing square will answer in nearly every case. One 
edge of each blade *\iOuld be perfectly straight and the 
edge of l should ee cut out in several places to see the 
blade e, when placed under the square. The two blades 
should be fastened together by a thumb-screvv. There 











66 


THE STEEL SQUARE 


should be three holes in l, one near each end and one in 
the middle, and a notch filed by each hole, so that the 
blade e, may be shifted when necessary. 



*To Find the Diagonal of a Square by this instrument, set 
the blade e to 8^ inches on the tongue and 12^ inches 
on the blade. Then screw the bevel fast; and supposing 
the side of the square in question is 11 inches, move blade 
e to the 11 inch mark on the tongue, keeping blade l 
against the square, when blade e will touch 15 9-16 inches 
on the blade, which is the required diagonal. There is no 
special reason for using 8^ and 12^6 ; other numbers may 
be employed provided the proportion of 70 to 99 exists 
between them. In the problem just solved as in all that 
follow, the bevel being once set to solve a particular ques* 


' J. O. Connell. 


































AND ITS USES. 


67 

tion will solve all the others of the same kind, till the bevel 
is altered. 


Polygons Inscribed in Circles. —In the following table, set 
the bevel to the pair of numbers under the polygon to be 
inscribed. 


No. 0/ sides. 

Radius. 

Side. 


3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

56 

70 

74 

Side 

60 

98 

22 

89 

80 

85 

97 

99 

87 

equal to 

52 

75 

15 

55 

45 

44 


radius. 


If we require the radius of a circle which will circum¬ 
scribe an octagon 8 inches on a side, we refer to column 8, 
take 98 parts on the blade and 75 on tongue, and 
tighten the bevel. As the side of a hexagon equals the 
radius of its circle, the side of an octagon must be less than 
the radius; hence we shift to 8 inches, that end of the bevel 
blade which gives the lesser number, in this case, on the 
tongue of the square, as the 75 parts to which the bevel 
was set are less than the 98. The required radius is then 
indicated on the blade. 

We will now explain the figures used in stepping round 
a circle forming inscribed polygons from three to twelve 
sides : Set bevel or fence to 12 on blade, and the number 
opposite each polygon on tongue; move to diameter of 
circle; answer of the side of polygon on tongue. 


Names. 

Triangle. 

Square. 

Pentagon. 

Hexagon. 

Heptagon.. 

Octagon. 

Nonagon.. 

Decagon. 

Undecagon... 
Dodecagon. 


No. 0/ Sides. 

. 3 

. 4 

5 

.6 

..... 7 
.8 

- 9 

.10 

.11 

.12 


Gauge Points 

io - 4o 
8-49 
7‘05 
600 

5'2I 

4 - 6 o 

4 11 
3 ' 7 * 

3'39 

J’ll 














6 $ 


THE STEEL SQUARE 


To divide a circle into a given number of parts, multiply 
the corresponding number in column one and the product 
is the chord to lay off on circumference. The side of a 
polygon is known, to find the radius of a circle that will 
circumscribe: Multiply the given side by the correspond¬ 
ing number opposite of polygon in column two. 


No. of 
Sides. 

Name of Polygon. A ngle. 

A ngle of 
Polygon, 

Column 1. 

Column 2. 

3 

Triangle. 


60 

i '732 

*5773 

4 

Square.. 

.90 

90 

1-414 

•7071 

5 

Pentagon .... 

.72 

108 

i*775 

•8510 

6 

Hexagon. 


120 

Radius. 

Side. 

7 

Heptagon.... 

.5i 3*7 

128 4-7 

•8677 

1-152 

8 

Octagon. 

.45 

135 

*7653 

1-3071 

9 

Nonagon .... 

.40 

140 

■6840 

1-4863 

10 

Decagon .... 

. 36 

144 

•6180 

i"6i8i 

11 

Undecagon... 

.3 2 8-11 

147 3* 11 

*5634 

**7754 

12 

Dodecagon... 

.30 

150 

•5176 

1 *93 2 3 


The side of a polygon is known, to find the length of 
perpendicular : Set bevel or fence to the tabulated numbers 
below. Example: The side of an octagon is 12, set bevel 
to 23 on tongue, 27 11-16 on blade. Blade gives the 
answer. 

No. 0/ Sides. 34567 8 9 10 11 12 

Perpendicular.. 9 I 30 13 273-4 277-10 503-4 281-2 313-4 26 

Side of Polygon 31 1-5 a 351-4 15 26 23 37 181-2 301-2 14 

To Inscribe three Equal Circles in a circle of given diame¬ 
ter. Set to 6 % on tongue and 14 on blade. Move the bevel 
to the given diameter on the blade and the required 

diameter appears on the tongue. 

Four equal circles require a bevel of 2*91 and 14. 

The following also, is another use for the square and 
bevel combined. 

If a person is drawing a machine on a scale of 1^ inch 
lo the foot, he may simply lay a common rule under the 












AND ITS USES. 


69 


square, touching the 12 inch mark on the blade, and the 
1 inch mark on the tongue; he then possesses a con¬ 
trivance by which he may easily reduce from one scale to 
the other. For instance, if a piece of stick 2 3 ^ inches 
square is to go into the construction, the draughtsman finds 
the 9^ inch mark on the blade, that is 2^ inches back 
from the 12 inch mark, and measures square out to the 

rule. This distance is the reduced section of the stick. 

/ 

A straight mark, drawn on a table or a drawing board* 
serves as well as a rule. 

Conveyors’-shaft 5 inches in diameter, 12 feet long, pitch 
of flights 9 inches; make a posteboard template; multiply¬ 
ing the diameter by 3-1416 gives the base, and the 9 is the 
altitude. The paper would be 9 inches altitude, 15 71-100 
base; draw a line along shaft, place altitude or 9 inches along 
this line, scribe along the hypothenuse; this gives the spiral 
course of flight. This principle also teaches how to cut round 
sticks of straight timber by marking along base of template, 
take square on each end the same as taking a stick out of 
wind, before striking lines. 

The cuts for the edges of the pieces of a hexagonal hop¬ 
per are found by subtracting the width of one piece at the 
bottom, viz., the width of same at top, and taking the re¬ 
mainder on the tongue, and depth of side on blade. The 
tongue gives the cut. For the cut on the face of the sides, 
take 7-12 of the rise on the tongue, and the depth of side 
on the blade. The tongue gives the cut. The bevel for 
the top and bottom edges is found by taking the rise on 
the blade, and the run on the tongue; the latter gives the 
cut. 

To find the cut of an octagonal hopper for the face of 


7 ° 


THE STEEL SQUARE 


the board and also the edge, subtract the rise from the 
width of side; take the remainder on the tongue and width 
of side on blade; the tongue gives the cut. The edge of 
the stuff is to be square when applying the bevel. The 
bevel for the top and bottom edges of the sides is found by 
taking the rise on the blade, and run on the tongue, the 
latter gives the cut. This makes the edges horizontal. 
The edges are not to be beveled till the four sides * are 
cut. 

Tc lay off Angles of 60° and 30°. —Mark any number of 

inches, say 14, on an indefinite line. Place the blade 
against one extremity of this distance, and the 7 inch mark 
of the tongue at the other. The tongue then forms an 
angle of 60 0 with the indefinite line, and the blade an angle 
of 30°. 


To Find the Bevels and Width of Sides and Ends of a 
Square Hopper. —Fig. 50. The large square represents the 
upper edges of the hopper and the small one the lower 
edges, or base. The width of the sides and ends is found 
in this way: Take the run a b on the tongue, and the per¬ 
pendicular height a d on the blade. It is thus found in the 
same manner as the length of a brace. To find the cut 
for a butt joint, take width of side on blade and half the 
length of the base on tongue; the latter gives the cut. F01 
a mitre joint take width of side on the blade and perpen¬ 
dicular height on tongue; the latter gives the cut. 

For the cut across the sides of the boards, take the run 


AND ITS USES. 


7 1 


a b on the tongue, and the width of side on blade; the 
tongue gives the cut. The inside corners of the sides and 
ends are longer than the outside, so if a hopper is to be of 



Fig. 5<x 

a certain size, the lengths of ends and sides are to be meas¬ 
ured on the inside edge of each piece, and the bevels struck 
across the edges" to these marks. This is only in case of 
butt joints. Of course if the hopper is to be square, the 
thickness of the sides must be taken from the ends. 

If the top and bottom edges are to be horizontal, the 
bevel is thus found: Take the perpendicular height of hop¬ 
per on the blade and the run on the tongue, the latter gives 
both cuts. A hopper can be made by the above method 
by getting the outside dimensions at top and bottom, and 
the perpendicular height. 

In large hoppers pieces are put down along the corner* 













72 


THE STEEL SQUARE 


to strengthen them. The length, and the bevel to fit the 
corner are thus found: Suppose the top of hopper is 8 feet, 
and the bottom 18 inches square. Find the diagonals of 
each, subtract the one from the other, and half the re¬ 
mainder is the run for the corner piece. From the length of 
this run, /, and the rise, a b, we find the length of the corner 
piece. To find the bevel or backing, take on the blade the 
length of the corner piece and on the tongue the rise; the 
latter gives the bevel. Another method is to draw the line, 
/, to represent the seat of the corner piece, set off square 
with this the line m, of the same length as the run, a b. 
Then draw n o, which is the length of the corner piece. To 
find the backing, draw a line, /, anywhere across /, at right 
angles therewith, and at its intersection with line, /, strike 
a circle tangent to ?i o. From the point of intersection of 
the circle with /, draw lines to the extremities of p. The 
angle made by these lines is the bevel or backing. 

Another method generally employed for finding the 
bevels of hoppers is to bevel the top and bottom edges of 
the sides and ends to the angle they are to stand at, then to 
lay a bevel set to a mitre, or angle of 45°, on the beveled 
edge, and that will lay off a mitre joint, while a try-square 
will lay off a butt joint. An angle of 45 0 will mitre only 
those boxes with sides which are vertical and square with 
each other. 

When the sides and ends of a rectangular box or hopper 
are of the same width, that is, when sides and ends slope 
at equal angles, the bevels, either butt or mitre, are found 
as for square hoppers. 

When a hopper has the sides and ends of different 
widths, that is, when sides and ends stand at different angles 


AND ITS USES. 73 

both having the same rise, find the cuts foi each from its 
respective rise, run and width. 

Roofing. —Fig. 51. A hip-roof with two corners out of 
square is given an example, the dimensions of which are. 
width 15 feet, rise of roof 5 feet, length 30 feet on the 



shorter side, 33 feet on the longer. The timbers ad, cd. 
eg, eg, are the hip rafters; j j the jack rafters. The seats 
of each hip rafter should form a square, so that each pair 
of jack rafters, j j, for instance, may be cut of equal length. 

Lengths and Bevels of Hip-Rafters. —We will first con¬ 
sider those on the square end of the roof. In order to find 
their length, it is first necessary to obtain their run, which 
is found as follows : Take half the width of building on both 
blade and tongue, whence is obtained the length of seat 
from g to e, at the intersection of the dotted lines. By similar 
use of the square, this length with the rise of roof, gives 
the length of the hip-rafter. The lengths of all the rafters 


















































74 


THE STEEL SQUARE 




should be measured along the middle, as the dotted lines 
show. This is the full length; half the thickness of the 
ridge-pole is to be taken off, measured square back from 
the bevel. 

The bevel of the upper end of a hip-rafter is called the 
down bevel. It is always square with the lower end bevel, 
hence these bevels are found by the parts taken on the 
square to find the lengths of the hip-rafters. Another 
method is to take 17 inches on the blade and the number 
of inches of rise to the foot, that is, the rise in inches di¬ 
vided by half the width of roof in feet—on the tongue. The 
tongue gives the down bevel, the blade the lower end bevel. 
The reason for the foregoing is that when the hip-rafters are 
square with each other, the seat of the hip is the diagonal 
of a square whose side is half the width of building. The 
diagonal of a square with a 12 inch side is 17 inches nearly. 
So if the rise of roof in 1 foot is 6 inches, the rise of hip- 
rafter will be that only in 17 inches. The directions here 
given assume that the hip-rafter abuts the ridge-pole at right 
angles, but as the ground plan of the roof shows that they 
meet at an acute angle, another bevel must be considered, 
called the side bevel of the hip-rafters. Were there no slope 
to the roof, the bevel where they meet the ridge pole would 
bean angle of 45 0 , as the hips would be square with each 
other. When a pitch or slope is given, the hips depart from 
the right angle, and therefore the side bevels are always 
less than 45°. Take the length of hip on the blade, and 
its run on the tongue; the blade gives the cut. 

Backing of the hip-rafters. The backs of the hip-rafters 
must be beveled to lie even with the planes of the roof. 
This bevel must slope from the middle toward either side. 


AND ITS USES. 


55 


It is found by taking the length of hip on blade, and the 
rise of the roof on tongue. The latter gives the bevel. 

To find the lengths of the jack-rafters: Suppose there 
are to be four between the corner and the first common 
rafter; then there are five spaces, which, by dividing 7 foot 
6 inches by 5, are 1 foot 2 inches from centre to centre of 
jacks. The rise of roof, also divided by 5, gives 1 foot rise 
for the shortest rafter. The run is 1 foot 6 inches; as both 
rise and run are given, the length down and lower bevels 
are found therefrom. The next jack has double the 
rise, run and length of the first; the following one three 
times, and the fourth four times. All the measurements 
are to proceed on or from the middle lines of the jacks. 

The side bevel of all the jack-rafters is obtained by taking 
the length of a common rafter on the blade and its run on 
the tongue; the bevel on the blade gives the result. 



Fig. 52. 


Let us now consider the end of the building out of square? 
Fig. 5'2 illustrates the method of laying down the seats of 
the hips. To find the lengths of these hips, the lengths of 
the seats must be got by taking half the width of building 
on blade, and the distance from the end of the dotted line 
crossing the roof, to the corner on the tongue. The length 





7 6 


THE STEEL SQUARE 


of the seat so obtained taken on the square, with the rise of 
the roof, gives the length of the respective hip-rafter. 

The down and lower end bevels are found as in the pre¬ 
vious hip-rafters. To obtain each side bevel, add the dis¬ 
tance from the dotted line to the corner and the gain of 
the hip-rafter; take the sum on the blade, and half the width 
of building on the tongue; the latter gives the cut. 

The lengths, etc., of the jack-rafters on the side, are de¬ 
termined as at the square end of the roof; the side bevel 
being found by taking the length of a common rafter on 
the blade, and the distance from the dotted line to corner 
on the tongue. The latter showing the bevel. 

The lengths of jack-rafters on the end. Assuming there 
are to be four jacks between the corner and the centre in¬ 
cluded, half the length of the end of the roof must be di¬ 
vided by 5. One side of the roof being 3 feet longer than 
the other, we place 3 feet, on tongue, and 15 feet, the width 
of building, on the blade, and thus obtain the distance from 
corner to corner on the end of the roof. Half this divided 
by 5 gives the distance of the jacks apart. The distance 
from where the middle lines of the hips meet to the middle 
point of the end of the roof is also to be divided by 5, the 
quotient giving the run of the shortest rafter. The rise is 
the same as for the jacks on the square end. 

These rules give the full length of rafter, so that when 
hips come against a ridge-pole or jacks against a hip, half 
the thickness of pole or hip, squared back from their down 
bevels, must be taken off. 

Side bevels of these jacks are obtained by adding the 
distance from the dotted line to the corner to the gain of 
a common rafter in running that distance; take this on th« 


AND ITS USES. 


77 

blade, and half the width of building on the tongue. The 
blade gives the bevel. 

Trusses. —Fig. 53. A is the straining beam, b the brace, 
t the tie beam. Generally the brace has about one-third 
the length of tie beam for a run. From the rise and run 
find the length and lower end bevel of the brace. After 
marking the lower end bevel on the stick, add to it just 
what is cut out of the tie beam. The bevel of the upper 
end of the brace where it butts against the straining beam 
i ft found in the following manner. Take the length of the 



Fig. S3 . 


brace, or a proportional part, and mark it on the edge of a 
board; take half the rise of the brace on the tongue, lay it 
to one of these marks on the board, and move the blade 
till it touches the other mark on board. A line drawn 
along the tongue gives the bevel for both brace and strain¬ 
ing beam. The angle made between brace and strain¬ 
ing beam is thus bisected. Lay off the measurements from 
the outside of the timbers. Put a bolt where shown, with 
a washer under the head to fit the angle of straining 
beam and brace. 








7* 


THE STEEL SQUARE 


There are quite a number of methods of obtaining ap¬ 
proximate proportions of the diameter of circles to their 
circumferences. . The true proportion, or, as it is some¬ 
times expressed, “ the squaring of the circle,” is one of 
those feats, like the discovery of “ perpetual motion,” and is 
as far from being accomplished now as ever. At any rate, 
it makes but little difference at this time, to the operative 
mechanic, whether the circle can be squared or not, so 
long as he can get near enough to the truth to satisfy the 
requirements at hand satisfactorily; and to aid him in this, 
the following method is shown of obtaining the circum¬ 
ferences of circles when the diameter is given, by use of 
the square. Of course, as shown in the cut, the rule will 
apply to circles of any reasonable dimensions. 



Let a b, Fig. 54, be a straight line, or the straight edge 
of a board ; then apply the square as shown, placing the 
16-inch mark on the blade at c, and the 5-inch mark on 
the tongue at d. See that the junctions of the blade and 
tongue of the square with the line a b, are accurately 
placed, for on this depends the truth of the results. Now, 
suppose we wish to ascertain the circumference of a circle 









AND ITS USES. 


79 


whose diameter is 8 inches; commencing at the point, c, 
we space off the diameter, 8 inches, three times, on the line 
c o, as shown at 8" 8" 8"; then square down the line 8" f, 
then c f will be the circumference of a circle whose diam¬ 
eter is 8. It will be seen, by dotted lines in the cut, that 
the circumference equals the diagonal of a rectangle whose 
sides are respectively 24 and 7at inches; so that by adopt¬ 
ing these figures (24 and 732) it enables the operative to 
use the full length and capacity of the square. The 
better way, however, is to work from a basis of 16 and 5, 
and draw the lines, c o and a b, to considerable length, so 
that they may be made available for dimensions beyond 
the range of the square. Now, let us suppose an instance 
where the circumference of a circle is wanted, whose diam¬ 
eter is 10; we simply space off three tens, or thirty inches, 
on the line c o, which, in this case, is at k. Square down 
from k to r, and c R is the length sought. 

Now, to prove this, let us proceed as follows: Diam. = 
10 X 3’1416 = 31*4160, or nearly thirty-one inches and 
fifteen thirty-seconds of an inch. Now, if we measure c R, 
we will find that the distance is exactly 31-4160 inches, 
and is, therefore, the answer sought. It will be seen by 
these examples that the circumferences of circles may be 
easily obtained when the diameters are known. So, also, 
may the diameters be found when the circumferences are 
known, for by laying off the circumference on the line a b, 
as c d in Fig. 54, for instance, and then applying the 
square as there exhibited, and dividing the distance from 
the heel of the square to the point c into three equal parts. 
One of these parts is the diameter of the circle whose cir¬ 
cumference equals the distance from c to d. 


do 


THE STEEL SQUARE 


In my experience, I have frequently been asked how a 
mitre, or equal joint, could be laid off by using the 
square. 

The matter is so simple, that it was thought unnecessary 
to insert it in the first edition, but the many inquiries on 
the subject that have been received since the work was 
published, induces me to give a few examples of the man¬ 
ner in which advanced workmen generally accomplish 
this end. Let Fig. 55 represent an oblique angle formed 


1 



Fig. 55 - 


by two parallel boards. To obtain the joint, a, space off 
equal distances from the point 1 to 3, 3, then square over 
from the lines, R, r, keeping the heel of the square at the 
points, 3, 3. At the junction of the lines formed by the 
tongue of the square at o will be one point, and 1 will be 
the other by which the joint line, a, is defined. 

To find the line of juncture for an acute angle, we pro¬ 
ceed as follows: Fig. 56 represents two parallel boards; 
1 the extreme angle, 3, 3 equal distances from the angle 1 
and are the points where the heel of the square must rest 
to form the lines o, 3 ; o shows the junction of the lines 
formed by the blade of the square. Draw a line from o to 
1, and the line, a, formed, is the bevel required. 




AND ITS USES. 


81 


1 



Fig. 56. 


It will be seen, by these two examples, that the bevel of 
a junction at any angle may be obtained by this method. 

Sometimes, when estimating on work, it becomes neces¬ 
sary to get the length of braces and other timbers, that 
would reauire considerable figuring to obtain if the usual 



method of finding the length of the third side of a right- 
angled triangle was adopted. The square, at this juncture, 
may be made use of with advantage, where the length of, 
the lines wanted is within the range of the instrument, and 
almost any dimensions may be manipulated, by making the 
subdivisions of the inch represent inches, feet, or yards. 
Suppose we want to get the length of a brace with unequal 
run of 7 and 12 feet resDectivelv. Lay the two-foot rule 






THE STEEL SQUARE 


S2 

across the square, putting the end on 7 on the tongue, and 
cutting the 12-inch line on the blade; then, as shown in 
Fig. 57, we will have on the side of the rule a b, 13 feet 
11 inches, or say 14 feet, which is near enough for the 
estimator’s purpose, and if required for working purposes, 
the exact length and bevels may be obtained by careful 
measurement. 


4.ND ITS USES. 


83 


PART IV. 

Miscellaneous Rules and Memoranda. —The practical 
carpenter and joiner will frequently want to use the more 
elaborate methods of obtaining solutions where the prob¬ 
lems are complicated and various; and the following rule? 
are inserted in this work with a view of reaching some of 
the problems that appear to be beyond the range of the 
Steel Square without making such intricate combinations 
as would be sure to lead to confusion in ordinary hands. 

Hip -Roofs. —The principles to be determined in a hip¬ 
roof are seven; namely : 

1 st. The angle which a common rafter makes with the 
level of the top of the building; that is, the pitch of the roof. 

2nd. The angle which the hip-rafter makes with the level 
of the building. 

3d. The angles which the hip-rafter makes with The ad¬ 
joining sides of the roof. This is called the backing of 
the hip. 

4th. The .height of the roof, or the “ rise.” as it is called. 

5th. The lengths of the common rafters. 

6th. The lengths of the hip-rafters. 

7th. The distance between the centre line of the hip- 
rafter and the centre line of the first entire common rafter. 

The first, fourth, fifth and seventh are generally given, 
and from these the others may be found, as will be shown 
by the following illustrations: Let a b c d Fig. 58, be 


8 4 


THE STEEL SQUARE 


the plan of a roof. Draw g h parallel to the sides, a d, 
b c, and in the middle of the distance between them. 
From the points a, b, c, d, with any radius, describe the 
curves a b, a b, cutting the sides of the plan at a, b. From 
these points, with any radius, bisect the four angles of the 
plan at r, r, r, t, and from a, b, c, d, through the points, 



Fig. 58. 


r, r, r, r , draw the lines of the hip-rafters, a g, b g, c h, d h, 
cutting the ridge-line, g h, in g and h, and produce them 
indefinitely. The dotted lines, c e, df are the seats of the 
last entire common rafters. Through any point in the 
ridge-line, 1, draw e i f at right angles to g h. Make 1 K 
equal to the height or rise of roof, and join e k, f k; then 
e k is the length of a common rafter. Make g o, ho, 
equal to 1 k, the rise of the roof, and join a o, b o, c o, d o, 
for the length of the hip-rafters. If the triangles, a^g,b<?g, 
be turned round their seats, a g, b g, until their perpen¬ 
diculars are perpendicular to the plane of the plan, the 
points, o o, and the lines, g o, g o, will coincide, and the 
rafters, a o, b 0, be in their true positions. 









AND ITS USES. 85 

If the roof is irregular, and it is required to keep ft* 
ridge level, we proceed as shown in Fig. 59. 

Bisect the angles of two ends by the lines a b, b b, c 
d g, in the same manner as in Fig. 58; and througl o 
draw the lines G e, g f, parallel to the sides, c b, d a, - 



spectively cutting a b, b b, in e and f; join e f; then the 
triangle, e g f, is a flat, and the remaining triangle and 
trapeziums are the inclined sides. Join g b, and draw h i 
perpendicular to it; at the points m and n, where H 1 cuts 
the lines g e, g f, draw m k, n l perpendicular to h i, and 
make them equal to the rise; then draw h k, i l for the 
lengths of the common rafters. At e, set up e m perpen¬ 
dicular to b e; make it equal to m k or n l, and join B in 
for the length of the hip-rafter, and proceed in the same 
manner to obtain Am, c m, d m. 

To find the backing of a hip-rafter, when the plan is 











86 


THE STEEL SQUARE 


right-angled, we proceed as shown in Fig. 60. Let Bb, bo. 
be the common rafters, a d the width of the roof, and a b 
equal to one-half the width. Bisect b c m a, and join a a y 
p a. From a set off a c y a d equal to the height of the 



roof a b, and join a d, d^; then a -d, d c are the hip- 
rafters. To find the backing: from any point h in a d, 
draw the perpendicular h g, cutting a a in g; and through 
ffdraw perpendicular to a a the line e f cutting a b, a d 
in e and f. Make gi equal to g h, and join k e, kf; the 
angle e kf is the angle of the backing of the hip-rafter c. 

Fig. 61 shows the method of obtaining the backing of 
the hip where the plan is not right angled. 

Bisect a d in a, and from a describe the semicircle 
A b D ; draw a b parallel to the sides a b, d c, and join 
A b, D b } for the seat of the hip-rafters. From b set off on 







AND ITS USES. 



Fig. 61. 

Fig. 62 shows how to find the shoulder of purlins: 

First, where the purlin has one of its faces in the plane 
of the roof, as at e. From c as a centre, with any radius, 
describe the arc dg ; and from the opposite extremities of 
the diameter, draw d h, g m perpendicular to b c. From 
e and f where the upper adjacent sides of the purlin pro¬ 
duced cut the curve, draw e i, f l parallel to d h, g m\ also 
draw c k parallel to d h % From / and i draw l m and i h 


87 

i 6 s the lengths b d, b <?, equal to the height of the roof 
b and join a <?, d d, for the lengths of the hip-rafters. To 
find the backing of the rafter:—In a *, take any point k, 
and draw k h perpendicular to a e. Through h draw/ h g 
perpendicular to a b, meeting a b, a d in f and g. Make 

/ equal to h k, and join //, g /; the fig is the backing 
the hip. 















Fig. 63. 






































AND ITS USES. 


89 


parallel to b c, and join k h, k m. Then ck 7 n is the down 
bevel of the purlin, and c k h is its side bevel. 

When the purlin has two of its sides parallel to the 
horizon. This simple case is shown worked out at f. It 
requires no explanation. 

When the sides of the purlin make various angles with 
the horizon. Fig. 63 shows the application of the method 
described in Fig. 62 to these cases. 

It sometimes happens, particularly in railroad buildings, 
that the carpenter is called upon to pierce a circular ol 
conical roof with a saddle roof, and to accomplish this 



Fig. 64. 


economically is often the result of much labor and per¬ 
plexity if a correct method is not at hand. 

The following method, shown in Fig. 64, is an excellent 













9° 


THE STEEL SQUARE 


one, and will no doubt be found useful in cases such as 
mentioned. 

Let d h, f h be the common rafters of the conical roof, 
and k l, i l the common rafters of the smaller roof, both 
of the same pitch. On g h set up g e equal to m l. the 
height of the lesser roof, and draw e d parallel to d f, 
and from d draw c d perpendicular to D f. The triangle 
d d c, will then by construction be equal to the triangle 
k l m, and will give the seat and the length and pitch of 
the common rafter of the smaller roof B. Divide the lines 
of the seats in both figures, d c, k m, into the same number 
of equal parts; and through the points of division in e, 
from g as a centre, describe the curves c a, 2 g, i /, and 
through those in b, draw the lines 3 /, 4 £, m a, parallel to 
the sides of the roof, and intersecting the curves in f g a. 
Through these points trace the curves c f g a , a f g a , 
which give the lines of intersection of the two roofs. Then 
to find the valley rafters, join c a, a a; and on a erect the 
lines a b, a b perpendicular to c a and a a, and make them 
respectively equal to m l ; then c b, a b is the length of the 
valley rafter, very nearly. 

Fig. 65 shows how a curved hip-rafter may be obtained. 
The rafter shown in this instance is ogee in shape, but it 
makes no difference what shape the common rafter may 
be, the proper shape and length of hip may be obtained by 
this method. It will be noticed that one side of the example 
shown is wider than the other; this is to show that the rule 
will work correctly where the sides are unequal in width, 
as well as where they are equal. Let a b c, f e c repre¬ 
sent the plan of the roof, fcg the profile of the wide 
side of the rafter. First, divide this rafter g c into any 


AND ITS USES. 


9 1 


number of parts—in this case six. Transfer these points to 
the mitre line e b, or, what is the same, the line in the plan 
representing the hip rafter. From the points thus estab¬ 
lished in e b, erect perpendiculars indefinitely. With the 
dividers take the distance from the points in the line F c, 



measuring to the points in the profile g c, and set the same 
off on corresponding lines, measuring from e b, thus estab j 
lishing the points 1, 2, 3, etc.; then a line traced through 
these points will be the required hip rafter. 

For the common rafter on the narrow side, continue the 
lines from e b parallel with the lines of the plan d e and 
a b. Draw a d at right angles to these lines. With the 
dividers as before, measuring from f c to the points in 
G c, set off corresponding distances from a d, thus estab- 
























9 2 


the steel square 


lishing the point? shown between a and h. A line traced 
through the porits thus obtained will be the line of the 
rafter on the rarrow side. This is supposed to be the 
return roof of ?> veranda, but is only shown as an example, 
for it is not customary to build verandas nowadays with an 
ogee roof, but with a rafter having a depression or cove in 
it. For accuracy it would be as well to make nearly twice 



the number of divisions shown from i to 6, as are there 
represented. 

It has been shown, in the forepart of this work, how the 
bevels and lines for hoppers may be obtained by the aid of 




































































AND ITS USES. 


93 


the square, and it is now proposed to show how the same 
results may be obtained by a system of lines. This 
method, in many shapes and forms, has been used from 
time immemorial by workmen, more particularly by car¬ 
riage makers to obtain the bevels of splayed seats; the 
present way of expressing it, however, is comparatively 
recent. 

If we make a i, Fig. 66, represent the elevation of our 
hopper, and b i a portion of the plan, we proceed as 
follows: Lay off n s, which is the bevel of one side, and 
n s p o the section of one end. 

Place one foot of the dividers at n, and with ns as 
radius describe the arc s u, intersecting the right line n u 
in the point u. At s erect the perpendicular s t, and draw 
the line u t at right angles to n u. Connect n and t ; 
then the triangle m n t is the end bevel required. The 
line n t is the hypothenuse of a right-angled triangle, of 
which n u may may be taken for the perpendicular and 
u t for the base. To find the mitre of which d e is the 
plan, project s and p, as indicated in the plan by the full 
lines. With s p as radius and s as centre, describe the arc 
p r. In the plan draw d g, on which lay off the distance 
s R, measuring from f, as shown by f g. Then G H f is 
the mitre sought. 

Fig. 67 shows the' rule for finding the bevels for the 
sides of the hopper. From m, the point at which e m im 
tersects b c, or the inner face of the hopper, erect the per¬ 
pendicular m l, intersecting r f, or the upper edge of the 
hopper, in the point l. Then l c shows how much longer 
the inside edge is required to be than the outside. In the 
plan draw t v parallel to s x, making the distance between 


94 


THE STEEL SQUARE 


the two lines equal to c f of the elevation, or, equal to the 
thickness of one side. From the point l in the elevation 



drop the line l w, producing it until it cuts the mitre lint 
n o, as shown at w. From w, at right angles to L w, 
erect the perpendicular w v, meeting the line t v in the 
point v. Connect v and u; then t v u will.be the angle 
sought. This bevel may be found at once by laying off 
the thickness of the side from the line e m, as shown by 
n p in the elevation, and applying the bevel as shown. 
This course does away with the plan entirely, provided 
both sides have the same inclination. 


















































AND ITS USES. 


95 


There are several other ways by which the same results 
may be obtained; some of these will no doubt occur to 
the reader when laying out the lines as shown here. 

Fig. 68 exhibits a method of obtaining the correct shape 
of a veneer for covering the splayed head of a gothic jamb. 



E shows the horizontal sill, e /the splay, /a the line of the 
inside of jamb, o the difference between front and back 
edges of jamb, b a the line of splay. At the point of junc¬ 
tion of the lines b a,/a, set one point of the compasses, 
and with the radius a b draw the outside curve of n ; then 
with the radius a s draw the inside curve, and n will be the 
veneer required. This will give the required shape for 
either side of the head. 










9 6 


THE STEEL SQUARE 


Sometimes the workman may have a semicircular groc-re 
to make, either as a pattern for casting or as a moulding of 
some sort. Now it is a well-known geometrical fact that 
the angle within a semicircular circumference is a right 
angle; this being admitted, it may be taken advantage of 
in making such hollow forms as I have mentioned, and 
which I illustrate by the Figure 69. The curve or serai- 



Fig. 69. 


circle may be worked out by a plane or other instrument, 
and to prove the correctness of the work, the steel square 
may be applied as shown. If the square touches at only 
two points, then the groove is not deep enough, or else it 
is too deep, in which case the operator must remove more 
stuff from the groove, or from the flat surface of the work, 
if permissible. By giving the square an oscillating motion, 
so as to make the corner sweep the entire surface of the 
curve, the accuracy of the work may be ascertained when¬ 
ever the square is applied. In this instance the square may 











AND ITS USES. 


97 


be used as a templet, and the advantages of it in such cases 
are that it may be used in any sized semicircular groove, 
thus saving a multiplication of templets. This principle may 
be extended further, as it is equally applicable to a hollow 
hemisphere as to a hollow semicylinder, which makes a 
knowledge of this fact of great value to wood or metal 
turners, as it can be easily seen that a hollow hemisphere 
must be nearly true if the corner or heel of the square 
touches every point, and the blade and tongue lie on the 
exact diameter. The operator can apply this principle to 
many cases, if he is at all ingenious. 

Here is a little problem that may come into use on many 
occasions: Suppose we want to cut a piece of sheet metal, 



Fig. 70. 


paper or thin wood so that it may form a cone, or a cover 
for a vessel of some sort; say the pitch or height at centre is 







•HE STEEL SQUARE 


98 

to be four inches, and the diameter of the base sixteen 
inches. Lay off the pitch on the tongue and half the 
diameter on the blade. The diagonal, from 4 to 8, as 
shown in Figure 79, gives half the diameter of the circle, to 
which set your compasses. Describe a circle as at Figure 
70, draw a line from the centre as shown, square off from a, 
on each side seven-elevenths of the diameter, ending at a 
and b. The gore left requires to be removed, then when 
the edges x x are brought together, the cone is complete. 

This solution may be used for many purposes. 

Sometimes it happens the workman will have heavy 
.-square balusters to cut on the angle or rake of the stair. 
The true angle can always be obtained by taking the height 
z>f riser on one blade and the width of tread on the other. 
Not only will this answer for cutting balusters either under 
the rail or on the string, but it will serve in every case 
where the angle or pitch is required. This is so apparent 
that an illustration seems unnecessary. 


AND ITS USES. 


99 


PART V. 

Sargent’s Sqaares are made of the best double- 
refined Steel and on improved machinery of delicate 
adjustment. The Squares are “true;” they.have the 
“hang” which makes them just right and they are 
accurately marked and nicely finished. Squares for all 
purposes and with different markings and labor saving 
tables and scales are made by this company and all genu 
ine Sargent’s Standard Steel Squares are stamped thus: 

All Squares bearing this stamp are 
fully warranted. 

Sargent’s Square No. ioo was used by the author 
to solve the problems in this book and he has no hesita¬ 
tion in saying that it is the most complete and decidedly 
the most convenient Square for the operative workman 
in the market. 

The description on the following pages show which 
tables and scales are used with the various Squares so 
that the reader will know'what number Square to ask 
for in the hardware store. 

Different Styles of Finish.— The finish of a Square 
is not only a matter of taste but of durability. The 

L OF C. 



100 


THE STEEL SQUARE 


SARGENT SQUARES are furnished in different styles, 
of which the following is a summary. 

POLISHED.—In these squares the metal is finely 
polished so that it does not easily rust or corrode, and if 
exposed to rain or damp the moisture may be thorough¬ 
ly removed from the surface by wiping with an oiled rag 
This cannot be done when the surface is in the least 
degree rough. 

NICKPiL PLATED.—Nickel plated Squares are 
not only very handsome, but are exceedingly durable. 
Nickel does not corrode readily; moisture and ordinary 
acids have no effect on it. A steel square of plain, 
polished steel if allowed to lie for a short time on an oak 
plank will become corroded and stained, leaving a dark 
mark on the wood. This does not occur when the surface 
of the steel has been nickel-plated. 

ELECTRO COPPER PLATED.—After being 
finely polished these squares are coated by an adhesive 
deposit of copper thrown down by electricity. This 
protects the surface of the metal and gives a very fine 
appearance to the square. The marking and lettering is 
also very distinct. 

ROYAL COPPER FINISH.—The royal copper 
finish is very handsome. It protects the steel very 
thoroughly and is very durable, not being liable to injury 
by any usage to which a fairly good mechanic would 
subject a square. The lettering and markings are very 
plain and easily read. 


AND ITS USES. 


IOI 


BLUED AND ENAMELED.—These squares 
are blued by a special process and the margins and 
figures are enameled either in white or yellow as may be 

s 

desired Sometimes this finish is referred to as gun metal. 

The royal copper and blued and enameled finishes 
make even the finest graduations easily read and the 
comparatively dull character of the general surface makes 
these squares easy on the eyes in outdoor work, even in 
the brightest sunshine. 

PATENT RAFTER TABLE, described on pages 
106 and 107. This table gives the length of rafters for 
any one of seven pitches of roof and for buildings of any 
width. 

BRACE MEASURE, described on page 109. This 
measure gives the length of common braces; its use will 
be easily understood by a reference to page 109. 

OCTAGON “ EIGHT-SQUARE ” SCALE, de¬ 
scribed on page no. This scale is used for laying off 
lines to cut an “eight-square” or octagon stick of timber 
from a square one. 

ESSEX BOARD MEASURE, described on page 
hi. This measure gives the contents in feet and inches 
of boards of various lengths and widths. 


102 


THE STEEL SQUARE 


POLISHED 

NICKEL 

PLATED 

Electro Copper 
Plated 

BRIGHT FINISH 

SIZE IN 
INCHES 

' 

W 

F-. 

<D 

rO 

a 

£ 

,£3 

o 

3 

© 

© 

o 

’£ 

Ah 

m 

u 

© 

rO 

a 

S3 

fc 

A 

o 

3 

<u 

© 

o 

•rH 

Oh 

w 

s. 

© 

a 

S3 

£ 

a 

o 

ci 

© 

© 

o 

•rH 

u 

Dh 

>> 

o 

CQ 

® 

S3 

be 

S3 

O 

I 

100 

1 

2 

2% 

3 

4 

5 

6 

7 

8 

9 

13 

14 

$3.33 

2.75 

2.50 

2.33 

2.30 

2.20 

2.12 

2.04 

1.96 

2.00 

1.83 

1.96 

1.92 

200 

101 

102 

102% 

103 

104 

105 

106 

107 

108 
109 

113 

114 

$4.17 

3.50 
3.17 
3.00 
2.92 
2.83 
2.75 
2.67 
2.58 
2.62 
2.46 
2.54 

2.50 

loop 

1 p 

2 P 

2 %P 

3 P 

$4.17 

3.50 

3.17 

3.00 

2.92 

24x2 

24x2 

24x2 

24x2 

24x2 

24x2 

24x2 

24x2 

24x2 

24x1 }4 

24x1 K 

24x2 

24x2 

16X.K 

16X1K 

16X1K 

16x1^ 

16x1 K 

16x1^ 

16x1 'A 

16x1 K 

I6xi>4 

16x1 

16xl 

16X1^ 

16x1^ 













14 P 

2.50 

15 

25.00 

115 

27.10 



24X3 

18x1^ 



16 

17 

6.25 

7.50 

116 

117 

7.08 

9.17 



30x2 

36x2 

* 1 - 

24x1^ 

30x1^ 





10 

11 

12 

52 

1.67 

1.58 

1.92 

2.08 

110 

111 

112 

152 

2.17 

2 08 
2.42 
2.58 



12X1J4 

12x1 

12X1^ 

12X1^ 

8x1 . 
8x1 

8X1 

8x1 



12 P 

2.42 



40 

41 

1.25 

1.17 

140 

141 

1.42 

1 33 



6X1 

6xl 

- 

4x^ 
4X3/ . 





























































































AND IIS USES. 


io 3 








HOW 

MARKED 






n of 

the letters see page 104. 







% 




Face 



xiack 


TWO FEET SQUARES 

For explanation of the 










-Brace Measure - Seepage 109 
Eight Square - - “ “ 110 


A 

1> 

c 

I) 

i; 

F 

fi 

ir 





vjr 

Essex Board Measure “ 111 

.. 

tf 

i 

¥ 

1 

TF 

1 J 
8 1 

r -12 & 
loo's 

fsV 

i 

T¥ 

1 

TO 

1 Brace Measure, 8 Square and 


TF 

1 

1 

¥ 

1 

1 

IF 

1 

If 

1-12 & 


A 

1 

1 


Jl 

1 

8 

1 

Essex Board Measure. 


T6 

8 

1 6 

8 

T¥ 

7 

l¥ 

7 

J 

• 

X 

F 

1 

¥ 

¥ 

¥ 

1 

1 2 

1 

8 

1 

T¥ 

1 

¥ 

Framing. 

* • 

l 

IF 

1 

7 

TF 

T 

T¥ 

7 

1 

1 2 

1 

7 

1 


1 

F 

1 

7 

1 

¥ 

1 

7 

T¥ 

7 

1 

T¥ 

1 

7 

! Brace Measure and 

Essex Board Measure. 

• • 

1 

¥ 

1 

T 

1 

¥ 

1 

7 

T¥ 

1 

2 

1 

T2 

T 


i 

¥ 

i 

7 

l 

8 

1 

7 

1 

7 

1 

2 

1 

7 

7 

J 

• • 

1 

¥ 

1 

7 

¥ 

1 

7 

7 

1 in. 

1 

7 

1 

7 

Essex Board Measure. 

• • 

1 

8 

1 

7 

¥ 

1 

7 

T¥ 

1 

7 

T¥ 

1 

7 


• . 

1 

8 

I 

1 

¥ 

1 

7 

1 

7 

1 

i 

1 

7 

| Brace Measure and 

L Essex Board Measure. 


1 

i 

1 

¥ 

1 

T 

1 

7 


1 

7 

• 

l 

¥ 

1 

T 

1 

¥ 

i 

7 

1 

7 

1 in 

1 

7 

1 

7 

Essex Board Measure. 










BRIDGE BUILDERS’ SQUARES. 


1 

i 

i 

1 

1 

1 

1 

l 

Body is 3 inches wide with a 1 inch 


1 F 

¥ 

TF 

8 

TF 

¥ 

T¥ 

¥ 

slot through the centre. Slot is 
marked in fths on both sides. 


1 

¥ 

1 

¥ 

l 

¥ 

1 

2 

1 

7 

1 in. 

i 

1 in. 

STONE CUTTERS’ 


i 

1 

¥ 

1 

¥ 

1 

2 

1 

7 

1 in. 

i 

7 

1 in. 

SQUARES, 


i 

1 

1 

1 

1 

i 

1 

i 

- 

• • 

¥ 

7 

8 

7 

T2 

7 

T2 

7 


• 

1 

¥ 

i 

8 

• • • • 

7 

7 

i 

• • • • 


• • 

TF 

i 

¥ 

TF 

t 

T*¥ 

i 

1 

T¥ 

1 

8 

12 INCH SQUARES. 

• • 

IF 

1 

¥ 

TF 

i 

T¥ 

¥¥ 

1 

T¥ i 

1 

TF 



F*F 

¥ 

fV 

i 

¥ 

tV 

i 

8 

1 

T¥ 

1 

¥ 

6 INCH 

• 

IF 

i 

TF 

1 

¥ 

T¥ 

1 

¥ 

1 

T¥ 

1 

¥ 

MACHINISTS’ SQUARES. 

• 














































































































104 


THE STEEL SQUARE 


Explanation of the Letters used in these pages 
to show how Sargent’s Squares are marked : 


Face of 
Square: 


A, Face of Body, Outside. 

B, “ “ “ Inside. 

C, Face of Tongue, Outside. 

D, “ “ “ Inside. 


Back of 
Square : 




Back of Body, Outside. 

“ “ “ Inside. 

Back of Tongue, Outside. 
“ “ “ Inside. 


The “ Face ” of the Square is the side upon which 
the name “Sargent & Co.” is stamped. 

The reverse side is the “Back.” The larger arm is the 
“ Body ” or “ Blade” and the shorter arm the “ Tongue.” 

Squares with 18-inch Tongue of the following num¬ 
bers furnished to order: Nos. 1,2, 3, 13, 14, 100, 101, 
102, 103, 200. 


Royal Copper Steel Squares. 


The following Squares can be furnished, at extra 
price, in Royal Copper Finish: 

No. 100 € ? Same Marks and Scales as No. 100 
No 100 CR, “ “ “ “ “ “ 100 R 






Blued Steel Squares. 

The following Squares can be furnished, at extra price, 
in blued steel with enameled figures and marks: 


Blued. 

White Enameled 
Figures and 
Marks 

Blued. 

Yellow Enameled 
Figures and 
Marks 

Same Marks 
and Scales as 
Numbers. 

IOO B 

IOO YB 

100 

1 B 

1 YB 

1 

2 B 

2 YB 

2 

2 H B 

2% YB 

2J 

3 B 

3 YB 

3 

12 B 


12 

14 B 

14 YB 

14 

IOO BR 

IOO YBR 

100 R 

1 BR 

1 YBR 

1 R 

3 BR 

3 YBR 

3 R 


/ 


Steel Squares with Metric Measure. 


Numbers 

Each 

Description 

Size in 

Inches 

Body 

Tongue 

314 

322 

$2.12 

1.00 

Steel Squares, Tapered.... 
Steel Squares, Not Tapered 

24x2 

24X1K 

T“< T-H 

X X 

CO Oi 

t-H t-H 


IIOW MARKED 



Face of Square 

Back of Square 


A 

B 

C 

D 

E 

F 

G 

H 

314 

322 

l A 

X 

A 

X 

% 

>4 

Centi- 
< ( 

metres 

i ( 

Centi- 

i 4 

metres 

( 4 


105 







































Squares 
with Rafter 
, Table. 




I 

l 




The Run of a rafter set up in place, is the 
horizontal measure from the extreme end of the 
foot to a plumb line from the ridge end. From 
A to B. 


C 



The Rise is the distance from the top of the 
ridge end of the rafter to the level of the foot. 
From C to D. 



The Pitch is the proportion that the rise 
bears to the whole width of the building. The 
above illustration shows J pitch ; the rise of 8 feet 
being ^ of the width of the building. 



The Cuts or angles of a rafter are obtained by 
applying the square so that the 12 inch mark on 
the body and the mark on the tongue that repre¬ 
sents the rise shall both be at the edge of the rafter 
The illustration shows 8 foot rise, the line A the 
cut for the foot end of rafter and B the cut for 
ridge end. 


"njn 

[T 

T 

T 

1 

TT 

TTTTTT 

■S 

2 >: 

1 




i 



w 

— 

«e 


RAFTERS 



— 



FEET-INCHES 




m 

or 


A TWELFTH8 


33 










o 

o 

l 

CD 

1 

* 

T 

ro 

[ 

o 

1 

CD 

'| J 

o> 

1 

4* 

RISE 


s 

S 

S 

K3 

S 


> 

__ "0 
-A3 h 

O 

— 

03 

03 

ro 

ro 

ro 

ro 

ro 


2 



ro 

vO 

*4 

4* 

ro 

- 



03 

CP 

- 

03 

- 

O 

4k 


— 

— 

cn 

4* 

4k 

03 

03 

03 

03 

03 



4» 

o 

ro 

O 

-4 

4^ 

- 


— 

— 

1 

CD 

"1 

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03 

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— 

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CD 

CD 

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— 








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cn 

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— 


o 

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03 

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•si 

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— 

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<r> 


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- 




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—i 

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CD 

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3 

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-4 


— 

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to 

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» 

03 



© 










— 


106 




























































































Rafter Table Directions. 


The Rafter Table includes the outside edge graduations of the 
back of the square on both body and tongue, and is in twelfths. 
The inch marks may represent inches or feet, and the twelfth 
marks may represent twelfths of an inch or twelfths of a foot 
(that is, inches) as a scale. The edge graduation figures above 
the table represent the “run” of the rafter, and under the proper 
figure on the line representing the “pitch,” will be found in the 
Table, the rafter length required. The “pitch” is represented by 
the figures at the left of the table and in the illustration under 
the word Pitch. 



12 

feet 

run to 4 feet rise is 

1/6 pitch. 


12 

u 

“ 6 “ 

U 

1/4 “ 


12 

u 

“ 8 “ 

a 

i /3 “ 


‘ 12 

i( 

“ 10 “ 

(( 

5/12 “ 


12 

u 

« 12 “ 

a 

1/2 “ 


12 

u 

“ 15 “ 

a 

5/8 “ 


12 

a 

“ 18 “ 

u 

3/4 “ 

) 

TO FIND 

THE LENGTH 

OF 

A RAFTER. 

For 

a roof with 

1/6 pitch (or the 

“rise’ 

’ 1/6 the width of the 

building) and 

having a “run” of 12 

feet 

follow in the Rafter 

Table 

the upper or 

1/6 pitch ruling, 

find, 

under the graduation 


figure 12, the rafter length required, which is 12 7 10, or 12 feet 
7 & 10/12 inches. 

If the “rim” is n feet, and the “pitch” 1/2 (or the “rise” 1/2 
the width of the building)' then the rafter length will be 15 6 8, 
or 15 feet 6 & 8/12 inches. If the “run” is 25 feet, add the rafter 
length for “run” of 23 feet to the rafter length for “run” of 2 

feet. 

When the “run” is in inches, then in the Rafter Table read 
inches and twelfths instead of feet and inches. For instance : if 
with 1/2 pitch the “run” is 12 feet 4 inches, add the rafter length- 
of 4 inches to that of 12 feet, as follows: 

For “run” of 12 feet the rafter length is 16 ft. 11 & 8/12 in. 

For “run” of 4 inches the rafter length is 5 & 8/12 in. 

Total.r.17 ft. 5 & 4/12 in. 

The “run” of 4 inches is found under the graduation “4” and 
is 5 7 11, which may be read 5 & 8/12 inches. If it were feet it 
would read 5 feet, 7 & 11/12 inches. 


107 










Sargent's Steel Squares 

with Rafter Table* ~ 


PATENTED JUNE 5, 1900. 


POLISHED 

NICKEL 

PLATED 

Electro Copper 
Plated. Bright Finish 

SIZE IN 

INCHES 

Numbers 

Price each 

Numbers 

Price each 

Numbers 

Price each 

Body 

1 

Tongue 

100 R 

$3 67 

200R 

$4 50 

100 PR 

$4 50 

24X2 . 

16X1^ 

1 R 

3 08 

101R 

3 83 

1 PR 

3 83 

24x2 

16 X 1 ^ 

3 R 

2 62 

103R 

3 25 

3 PR 

3 25 

24x2 

16 X 1 K 


HOW MARKED 

For explanation of the letters see page 104 . 




i FACE 

BACK 

Two Feet 

Squares 

A 

B 

C 

D 

E 

F 

G 

H 

100R1 











• 

f 

200 R 

- 

T6 

8 

A 

¥ 

tV 

3 V 

tV 

ro 


Brace Measure 

100 P R 











8 Square and 

1 R 











Patent Rafter 

101 R 


tV 

i 

iV 

8 

tV 

8 

tV 

8 1 


Table 

IPRj 












3R] 









1 Brace Measure 

103IU 

1 

IT 


iV 


tV 

i 

tV 

i 


and Patent 

3 PR 










1 

Rafter Table 


The patent Rafter Table gives the measure of the rafter 
for any one of seven pitches of roof, based upon the length of 
the horizontal measurement of the building from the cenire to 
the outside. Full description and directions for using may 
be found on pages 106 and 107. 


108 


























































m 


— 

T- 

— CM 


— CO 


=4 

— 

- 

— 10 

— 

— 

— vo 


— 

— CO 

- 

— ON 


0 



. « 

CN 

— 


CM 


CO — 


lO 


NO 


00 


ON 


O n 


— CO 


W 

U!l 

3 

I 1 I II | 1 1 I I I 

4 

4 2 


6 2 

3 

N 

CO 

G 

10 2 

9 

2.3 

3 / 

4 3 

6 / 

6 3 

9 




Brace Measure. 


This is along the centre of the back of the 
“tongue,” and gives the length of the com¬ 
mon braces. 

eo, 91-in the scale means, that if the 

run is 36 inches on the post, and the same 
on the beam, then the brace will be 50/91-100 
inches, as shown in the diagram at the right 
hand corner of this page. 

If the run is 51 inches on both beam and 
post, then the brace will be 72, 12-100 inches, 
and so on. 


E09 

















































































u 


Octagon 
Eight=Square 
Scale. 




04 


SI 



This scale is along the middle of the face of 
the tongue, and is used for laying off lines 
to cut an “eight square” or octagon stick of 
timber from a square one. 

Suppose the figure A, B, C, D, is the butt 
of a square stick of timber 6X6 inches. 
Through the centre draw the kites A B and 
C D, parallel with the sides and at right angles 
to each other. 


h A a 


H 

✓ 

/ 

E 

\ 

X 

\ 

\ 

\ 

X 

x 

X 

N 

X 

x^ 

G 

/ 

/ F 


1 I ■ WII I ( 

e B d 


With the dividers take as many spaces (6) 
from the scale as there are inches in the width 
of the stick, and lay off this space on either 
side of the point A, as A a and A h ; lay off 
in the same way the same space from the 
point B, as B d, B e; also C f, C g and D b, 
D c. Then draw the lines a b, c d, e f and 
g h. Cut off the solid angle -E, also F, G and 
H; this will leave an octagon, or “eight-square” 
stick. This is nearly exact. 


no 

































































































Essex 

Board 

Measure. 



la 


60 18 
60 84 88 24 30 



1 

100 




■' ' 1 EE'ET’"' E 1 1 

Jill 

JILL 

1111 

ITT rjTi11j1 ri 1 11 1 it| 


The figure 12 in the graduation marks on 
the outer edge represents a one-inch board 
12 inches wide and is the starting point for 
all calculations; the smaller figures under 
the 12 represent the length. 

A board 12 inches wide and 8 feet long 
measures 8 square feet, and so on down 
the table. Therefore to get the square feet 
of a board 8 feet long and 6 inches wide 
find the figure 8 in the scale under the 12 
inch graduation mark and pass the pencil 
along to the left on the same line to a point 
below the graduation mark 6 (representing 
the width of the board), and you stop on 
the scale at 4, which is four feet, the board 
measure required. If the board is the same 
length and 10 inches wide, look under the 
graduation mark 10 on a line with the fig¬ 
ure 8 before mentioned, and you find 6 and 
8-12 feet board measure. If 18 inches wide, 
then to the right under the graduation 
mark 18, and 12 feet is found to be the 
board measure. If 13 feet long and 7 
inches wide, find 13 in the scale under the 
12 inch graduation, and on the same line 
under the 7 inch graduation, will be found 
7 and 7-12 feet board measure. If the board 
is half this length, take half of this result; 
if double this length, then double the result. 
For stuff 2 inches thick double the figures. . 

In this way the scale covers all lengths of 
boards, the most common, from 8 feet to 15 
feet, being given. 












_ 



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— 


in 


























































































































SARGENT & COMPANY'S 

WOOD-BOTTOM 
AND IRON 


PLANES 



Good Workmen call for the best tools. In 
these planes we have the best material, properly 
put together, well proportioned and warranted 
in every way as the best Planes manufactured. 
The patent adjustment is perfect, and the steel 
cutters are tempered by an improved process, 
assuring the best results. Dealers are author¬ 
ized to fully warrant every Plane having this 
stamp upon the Steel Cutter. 








Books for Carpenters* 


We make a specialty of supplying books on carpentry 
and building work and we would be pleased to send you, 
free, our catalog “S,” as this describes some of the best 
books a carpenter can have that will aid him in mastering 
his trade. 

We have books for nearly every trade, and if you wish to 
know how to do something, write us about it and no doubt 
we will be able to recommend a book that will tell you all 
about it. 

We publish a monthly paper The Practical Carpenter at 50 
cents per year. Each issue contains at least 16 6x9-inch 
pages of reading matter of special interest to carpenters and 
builders. You ought to subscribe as it will keep you posted. 
Send for a free sample copy. 

Below we give a short list of some of the useful books we 
have. Any book sent prepaid on receipt of price and if the 
book don’t suit send it back and we will return your money. 

INDUSTRIAL PUBLICATION CO., 

16 THOMAS ST., NEW YORK. 


Roof Framing Made Easy, by Maginnis. A practical method 

for laying out various kinds of roofs. Simple and reliable; 
164 pages, 100 illustrations, octavo, handsomely bound in cloth 
for .$1.00 


Cloth Bound Books at 50c. each. 


Steel Squares and Their Uses, by Hodgson, gives many 
new problems. A companion volume to this book, 80 pages, 
illustrated. 

Practical Carpentry, a practical guide to laying out car¬ 
penter’s work, 144 pages, 300 illustrations. 

Hand Saws, their use, care and abuse. Tells how to set, 
file, etc., 96 pages, illustrated. 






Cloth Bound Books at 50c. each. 


Stair Building Made Easy. A simple guide for laying out 
. nd erecting different sorts of stairs, 160 pages, ioo illustra¬ 
tions. 

New System of Hand Railing. Simple, economical and 
may be learned in an hour. Illustrated by folding plates. 

How to Measure Wood Work for Buildings, by Maginnis, 

79 pages, 160 illustrations. 

Easy Lessons in Architecture, by Mitchell. A simple text 
book for the beginner; 94 pages, 100 fine illustrations. 

The Carpenters’ and Joiners’ Pocket Companion, by Mo¬ 
loney; contains rules, data and directions for laying out 
work; 102 pages. 

The Steel Square Pocket Book, by Stoddard. A handy and 
practical treatise on the square; 108 pages, 112 illustrations. 

The Workshop Companion, by Phin. A useful book of 
recipes for mechanics; 160 closely printed pages. 

How to Mix Paints, by Godfrey; 64 pages; illustrated. 

The Hardwood Finisher, 112 pages about staining, finishing, 
polishing, etc. 

Hints for Cabinet Makers and Furniture Men, 130 pages of 
useful recipes and hints. 

25c. Books in Paper Covers. 

Carpentry. 

Practical Joinery. 

Rafter and Brace Tables. 

Scaffolding. 

Hints and Aids in Building and Estimating. 

How to Become a Good Mechanic. 

Draughtmanship. 

The Slide Rule and How to Use It. 

Home Handicrafts. 

Hints for Painters. 

Concrete. 

Artificial Stone, Terra Cotta, etc. 

Masonry. 

Bricklaying. 

Plastering. 

Cements and Glue. 

Plumbing and Tinsmiths’ Work. 

Slating and Tiling for Roofs. 

Wood Turning. 

Glazing. 

The Lightning Calculator. 




Usef\il Instructive Books 

AT 25 CENTS EACH 

The books in the following list are practical and reliable, 
being especially prepared for self-instruction. 

The books are 12mo in size, printed on good paper, and 
well illustrated wherever the subject needs it. Bound in 
neat paper covers. The cheapest series of really good tech¬ 
nical books now in print. 

Books on other subjects in preparation. 

Any book sent postpaid on receipt of 25c., or five for one 
dollar. 

Industrial Publication Company 

16 Thomas Street, NEW YORK 


Carpentry ; A Practical Manual edited 
by John Black. 92 pages; illustrated by 
100 engravings. 

This book treats on the principles of the subject, 
the strains in the various members of framed struc¬ 
tures; joints of various kinds; simple roofs, king¬ 
post roofs, queen-post roofs, hip roofs, roofs cover¬ 
ing buildings of irregular plan. Mansard roofs; prop¬ 
erties of timber; flooring and timbers for supporting 
same; trussed beams; partitions, from simple exam¬ 
ples to elaborate examples of trussed partitions; 
method of deadening sound, etc., etc. 

Practical Joinery.— Edited by John 
Black. 92 pages; illustrated by 130 en¬ 
gravings. 

A book that points out the best methods in the 
various departments of joiner's work, such as mould¬ 
ings, classic and Gothic; tongue and groove joints, 
dowel joints, miter joints, lap joints, dove-tail joints, 
mortice and tenon joints; cutting timbers, making 
doors, paneling; hanging doors and windows; sky¬ 
lights; laying down floors; hardwood floors; con¬ 
struction of niches, etc. Woodworking machinery; 
picture frame making, and instructions on how to use the diagonal scale. 

The Steel Square.— By F. T. Hodgson. 

48 pages; illustrated by 38 engravings. 

This work is intended as an elementary introduction 
for the use of those who have not time to study the 
larger works on the same subject. The book shows how 
some difficult problems in carpentry and joinery are 
simplified and solved by the aid of the carpenter’s steel square, together with a full 
description of the tool, and explanations of the scales, lines and figures on the blade 
and tongue, and how to use them in everyday work. Showing how the Square may be 
used in obtaining the lengths and bevels of rafters., hips, groins, braces, brackets, 
purlins, collar beams and jack-rafters. Also its application in obtaining the bevels 
and cuts for hoppers, spring mouldings, octagons, diminished styles, etc. 










































2 


USEFUL INSTRUCTIVE BOOKS AT 25 c. EACH. 



Scaffolding. —Edited by John Black. 
90 pages; illustrated by 45 engravings. 

The subject of the erection of proper scaffolding 
for various purposes is practically treated in this book. 
A short history of ancient scaffolding is given, together 
with directions for erecting ordinary bricklayers’ scaf¬ 
folds, ladders; shoring and needling for supporting 
buildings ; cranes, lifts, hoists, traveling cranes, trans 
porters, cable railways; repairing steeples and tall 
chimneys; descriptions of various schemes and aims 
that will give a fund of practical information to every 
one called upon to erect scaffolding at the least cost 
in time and money. 


Hints and Aids in Building and Esti- 

mating. A hand-book for every one en¬ 
gaged in the erection and repair of build¬ 
ings. 36 pages. 

This useful little book gives hints and prices ; tells 
how to measure: explains building terms, together 
with a number of tables; schedule of architects’ 
charges and form for building contract; form for 
making estimates ; cost of doing work ; work a man 
will do ; diameter and height of chimneys ; weight of 
various roof coverings ; painting. 





Cements and Glue. —A practical treatise 
on the preparation and use of all kinds of cc 
ments, glue and paste. By John Phin, author 
of “ How to Use the Microscope.” 58 pages. 

Every mechanic and householder will find this volume of 
almost every-day use. It contains nearly 200 recipes ft r 
the preparation of cements for almost every conceivable 
purpose, amongst which are recipes for waterproof ce¬ 
ments. bottle cement, cap cement, fireproof cement, various 
glues, rubber cement, iron cement, ivory cement, leather cement; to cement glaSs, 
stone, etc., to metal; different kinds of paste, etc. 




Plastering. —Edited by John Black. 90 pages; 
illustrated by 40 engravings. 

This book contains quite a fund of information of a plain and 
practical character. A short history of the craft is given and 
then, in order, the materials and methods are described. The 
art of plastering is described, also the tools used, etc. Other 
chapters treat of various cements and their uses; lime and 
cement mortars ; methods of outside and inside work ; decora¬ 
tive plastering and details ; stucco coloring and plaster paint¬ 
ing; fixing tiles, mouldings, scagliola, fibrous plaster and other 
plasters ; making a scagliola column ; making mouldings in fibrous plaster; use of 
asbestos in the making of fireproof plaster, etc. 

Concrete. —By Frank Jay. 94 pages; 
illustrated by 38 engravings. 

The extensive use of concrete at the present time 
makes this little manual “ fill a long felt want.” - It 
is written by an expert of many years’ experience in 
concrete work. The various methods now in vogue 
are described. The following is a synopsis of the 
contents : Historical; materials ; aggregates and 
proportions ; building in concrete ; apparatus for 
erecting buildings ; floors, joists and wallings ; con¬ 
crete paving and flooring; methods of construction ; comparisons of different sys¬ 
tems ; fire resisting qualities; pavings in situ; moulds for concrete work; making 
artificial stone ; coloring and hardening ; armored concrete ; manufacture and use of 
concrete for dams, breakwaters, etc. ; use of concrete in building walls, piers, col¬ 
umns, floors, chimneys ; building armored concrete beams, etc., etc. 










































USEFUL INSTRUCTIVE BOOKS AT 25c. EACH. 


3 


Bricklaying. — Edited by John Black. 
88 pages; illustrated by 100 engravings. 

This book is not intended as a text-book but as a 
guide to the best practice. The instruction is given in 
simple, clear language, and the following points are 
treated: Classes and kinds of bricks; bonding for 
.’ouudations and walls; pilasters; piers; window open¬ 
ings; arches, their setting out and construction; 
bridges, their development and construction; flues, 
fireplaces and chimneys; oriels and bay windows; 
stops; quoins, cornices, gables; ornamental brickwork; 
fixing tiles ; general memoranda, etc. 




94 


Masonry. Edited by John Black. 
pages; illustrated by 80 engravings. 

This little book deals with the operations of masonry 
m a thoroughly practical manner, describing the vari¬ 
ous materials, the preparation of surfaces, building of 
all sorts ot walls and foundations of various kinds; 
varieties of masonry ^openings ; arches ; buttresses ; 
domes; vaults; together with a short history and sim¬ 
ple directions about drawing. The numerous illustra¬ 
tions show how stones are cut and dressed, showing 
ashlar, hammer or pick dressing, chisel dressing rub¬ 
bing ; chamfered stones; to produce a perfectly plane 
surface; winding surfaces, etc. F 

Tinsmiths' Work.— 


Plumbing and 

Edited by John Black. 92 pages; illustrat¬ 
ed by 80 engravings. 

This little book deals with roof-covering; roof work; 
gutters; covering flats, platforms, dormers, ridges, 
etc.; finials; pipes; the storage and supply of water; 
delivery and control of water; elementary sanitation; 
soil pipes, closets, baths, traps; lead lining for sinks; 
development of surfaces so as to get the cuts for el¬ 
bows, angles, etc.; how flashings should be put down ; 


snow boards 
water supply 




principles of water supply; drinking 
float tanks; hydraulics, etc- 

Slanting and Tiling. — Edited by John 
Black. 93 pages; illustrated by 50 engrav¬ 
ings. • 

Many buildings now have slate or tile roofs, and 
this book gives concise information about the various 
points, such as a history of many roof coverings from 
the earliest times, also of modern roof coverings, tiles, 
slates and slating ; tools used by roofers ; preparation 
of the roof; different kinds of roofs; description of 
various forms of framing ; more about tiling ; how to 
lay slates; copper and lead roofs; soldering irons, 
etc.; zinc roofs and concrete roofs ; the use of expand¬ 
ed metal, etc. ^comparison of different roofing mater¬ 
measuring up slating, tiles and other roof covering, with re- 


ials; thatched roofs , 
marks on different methods of measuring. 






Decorating- —Edited byjNO. Black. 

95 pages; illustrated by 15 engravings. 

The subjects treated in this practical book are 
as follow: Theory of colors; color blindness; 
the decoration of churches, libraries, dining, sit¬ 
ting and bedrooms; history and application of 
glass painting; stained glass in decoration; scene 
painting; carving in wood and stone; stencils; 
metal work in decoration, tile decoration, fres¬ 
coes, plasters; sanitary decoration of the house, 

etc. The hints contained in this book will be appreciated by the practical painter and 
decorator, as well as every householder who desires to have his home neatly and 

tastefully oraacaeated. 


^22529 





















































USEFUL INSTRUCTIVE BOOKS AT 25c. EACH 


Home Hak.ndicra.fts,— A Practical Guide 
for Amateurs. 92 pages; illustrated by 60 
engravings. 

This book will- be appreciated by every one who 
takes advantage of their spare hours to construct or 
decorate some detail of their homes. The following 
synopsis of the contents will show the wide range of 
subjects : Tools, materials, planing, workbench, mor¬ 
tise and tenon joint, halved joint, half lap joint, 
simple dooi'S, out buildings, kitchen table, saw horses, 
picture frame making ; gluing, lathing, whitewashing, 
paperhanging, painting, window boxes, soldering, table fountain, renewing sash lines, 
rustic railings, meat safes, bicycle racks, green-house work, constructing sun dials, 
making a drawing-board, French polishing, kennels, dovecots, poultry houses, useful 
recipes, etc. 

?5he Slide Rule and How to Use It. 

30 pages. By F. T. Hodgson. 

This is a compilation of explanations, rules and in¬ 
structions suitable for mechanics and others interested 
in the industrial ai’ts. Rules are given for the measure¬ 
ment of all kinds of boards and planks, timber in the 
round or squai'e, glaziei's’ work and painting, brick¬ 
work, pavioi’s’ work, tiling and slating, the measux-e- 
ment of vessels of various shapes, the wedge, inclined 
planes, wheels and axles, levers, the weighing and 
measuring of metals and all solid bodies, cylindei-s, cones, globes, octagon rule and 
formulae, the measurement of cii’cles. a comparison of French and English measures, 
together with much information useful to carpenters, bricklayei*s, glaziers, paviors, 
machinists and other mechanics. 




T5he Engineer’s Slide Rule and Its Applications. —By Will¬ 
iam Tonkes. 35 pages. 

A complete investigation of the principles upon which the slide rule is constructed, 
together with its application to all the purposes of the practical mechanic, such as 
multiplication, division, exti-acting roots, powei'S of numbers, meashrement of vai’ious 
plane and solid figures, estimating the weight of various materials, ge. metrical prob- 
ems, propox'tion, change of gears for scx*ew-cutting, calculations on leve. 3, etc. Pos¬ 
sessed of either of the above books and a good slide rule, mechanics might carry in 
their pockets some hundreds of times the power of calculation that they now have in 
their heads, and the use of the instrument is very easily acquired. 



Drawing Instruments. — By An Old 

Draftsman. 48 pages; illustrated by 20 en¬ 
gravings. 

A treatise on drawing instruments, with rules for their 
use and care ; dividers, compasses, ruling pens, bow instru¬ 
ments; special forms of instruments; how to handle them; 
drawing boards, paper, tee-squares, triangles, curves, 
scales, thumb-tacks, ti'acing paper and cloth, inks, pencils, 
protractors ; useful memoranda and data for every owner 
of di'awing instruments. 


Painting and Varnishing.— Edited by 
John Black. 94 pages; illustrated by 20 
engravings. 

A pi*actical manual treating on materials, principles 
of color, mixing paints, pi*epai'ation of surfaces, dis¬ 
temper, decoration, hints ou dealing with customers, 
hints on the use of bi’ushes, what colors to use in dif¬ 
ferent rooms, etc.; varnishes and vaimishing ; recipes 
for making various kinds of varnish; graining, grain- 
ing to imitate oak, mahogany, satin wood, walnut and 
x'osewood ; marbling, white veined, dove colored, red, 
green, jasper, black and gold, Florentine, Sienna, etc.; inside painting and decoration 
gilding, sign writing, lettering, alphabets, stencils, monograms, etc.? outside painting 
brickwork, metal work, etc- $ 

































USEFUL INSTRUCTIVE BOOKS AT 25c. EACH. 


S 



Hints for Painters, Decorators and 

Paper Hangers. — Prepared with Special 
Reference to the Wants of Amateurs. By 
An Old Hand. 60 pages. 

A most useful book treating on the preparation of 
sui'faces, materials used as bases and vehicles, white 
lead, linseed and other oils, driers, coloring paints, 
mixed paints, operations, taste in color, general re¬ 
marks on graining, miscellaneous receipts, paper¬ 
hanging, cleaning paper-hangings, varnishing paper, 
making paste; useful hints, tables, etc., for estimating cost of work aud materials 
which will prove of great value to the beginner in the painting business. 


Success with Recipes. — A practi¬ 
cal guide to success in the use of recipes, 
formulae, etc., with hints on chemical 
and mechanical manipulation. Intended 
as a supplement to all books of recipes. 
By John Phin. 44 pages. 

While it is an undoubted fact that many of the 
recipes published in the ordinary collections are 
erroneous, either from original blunders on the 
part of the authors, or from mistakes in copy¬ 
ing, failure in the use of others frequently arises from defective information and 
vicious methods on the part of those who attempt to put them into practice. The ob¬ 
ject of the present book is to give such hints aud cautions as will enable the worker to 
secure success where success is possible, and where the products are intended for sale 
it gives valuable advice as to the best methods of putting them on the market. 



Useful a.nd Precious Minerals, —How to 

find them; how to test them and how to estimate 
their value by simple methods and easily ob¬ 
tained appliancps. Intended for the use of non¬ 
experts. Edited by John Phin. 72 pages; illus¬ 
trated by 4 engravings. 

This book was prepared to meet the wants of the non¬ 
expert so that they may, by simple tests, ’ know if their 
‘•find” is valuable or only useless dirt. To this end the 
book gives general hints on the examination and testing of 
minerals; distinguishing characteristics of minerals; a sim¬ 
ple method of finding specific gravity; scale of hardness,’ 
malleability, color, luster, crystallization, chemical compo- 
sitiion; prospecting or searching for minerals, etc. 



How to Become a Good Mechanic. 

—Intended as a practical guide to self- 
taught men, telling what books to use; how 
to begin; what difficulties will be met; 
how to overcome them ; in a word, how to 
carry on such a course of self-instruction 
as will enable the young mechanic to rise 
from the bench to something higher. By 
John Phin. Second edition, revised and 
greatly enlarged. 68 pages. 

This book is not a text-book, but rather a guide to the use of these books. The 
author briefly outlines a course of study for mechanics who wish to advance them¬ 
selves. The notes and instructions given are of a kind ^ atap pea 1 d f ' I- |J' 1 , y a t°to m-eater 
sense and reason of the young student, and cannot fail to act as a stimulant to greater 
Sorts Ik obtafntat the Lowfodge sought The difficulties which may be exacted by 
the student are dwelt upon, and valuable suggestions as to the proi i 
overcoming them are given* 






















































6 


USEFUL INSTRUCTIVE BOOKS AT 25c. EACH. 


The Pistol as a Weapon of De- 

fence, in the House or on the Road. 
50 pages. 

This work aims to instruct peaceable and law- 
abiding citizens in the best means of protecting 
themselves from the attacks of the brutal and the 
lawless, and it is the only practical book published 
on this subject. Its contents are as follows : The pistol as a weapon of defence ; the 
carrying of firearms; different kinds of pistols in market; how to choose a pistol , 
amunition, different kinds; powder, caps, bullets, copper cartridges, etc.; best form 
of bullet; how to load ; best charge for pistols ; how to regulate the charge ; care of 
pistol; how to clean it; how to handle and carry it; how to learn to shoot; practical 
use of the pistol; how to protect yourself and disable your antagonist. 



Shooting on the Wing. —Plain direc¬ 
tions for acquiring the art of shooting on 
the wing. With useful hints concerning 
all that relates to guns and shooting, par¬ 
ticularly in regard to the art of loading so 
as to kill. To which has been added several 
valuable and hitherto secret recipes of very 
great practical importance to the sports¬ 
man. By An Old Gamekeeper. 88 pages; 
illustrated. 

This book tells how to choose the gun. about ammunition, gun cases, how to load 
the gun, how to clean it, how to handle and how to carry it, how to learn to shoot, 
finishing touches, useful hints, recipes and miscellaneous matter. The book contains 
a novel and most valuable feature found in no other work on this subject. This is a 
series of graduated lessons by which the self-taught young sportsman is enabled to 
advance step by step from such easy marks as a sheet of paper nailed on a fence to 
the most difficult trap-shooting and the sharpest snap-shots. 

What to Do in Case of Accident.— 

A book for everybody. 96 pages. 

This is one of the most useful books ever published 
It tells exactly what to do in case of accidents, such as, 
severe cuts, sprains, dislocations, broken bones, burns 
with fire, scalds, burns with corrosive chemicals, sun 
stroke, suffocation with foul air, hanging, drowning 
frost-bite, fainting, stings, bites, starvation, lightning 
poisons, accidents from machinery, gun-shot wounds 
etc., etc. It ought to be in every house and workshop 
for young and old are liable to accident, and the directions given in this book might 
be the means of saving many a valuable life. 






The Sun. —A familiar description of 
his phenomena. By Rev. Thomas Will¬ 
iam Webb. Author of “Celestial Objects 
for Common Telescopes.” 80 pages; illus¬ 
trated by 17 engravings. v 

A book for every one interested in Nature, as it 
simply and fully describes the sun, tells about spots, 
eclipses, etc., in a very attractive style, so that the 
ordinary reader who does not understand astronomy 
may thoroughly comprehend and enjoy the subject, 
a fair idea of the wonderful universe of which we are a part. 


Ar'A: 


V- 



A reading of this book will give 


Rhymes of Science; Wise and Otherwise. —By Oliver W 

Holmes, Bret Hart, Ingoldsby, Prof. Forbes, Prof. J. W. Mc.Q 
Rankine, Hon. R. W. Raymond and others. 66 pages ; illustrated 

A collection of scientific rhymes that will form pleasant reading for any one later 
ested in scien { 

















USEFUL INSTRUCTIVE BOOKS AT 25c. EACH. 


7 


Glazing. —Edited by John Black. 94 pages, 
illustrated by 50 engravings. 

The information in this manual will save much time and 
trouble to the practical glazier, while the apprentice will find 
it a trustworthy guide. Beginning with a history of the in¬ 
vention of glass and Its manufacture, the practical instruc¬ 
tion of glass cutting is taken up. Directions and hints are 
given for cutting oval, circular and irregular pieces, as well 
as rectangular ones. Practical notes are given on the plan¬ 
ning and placing of windows so as to get the best effect of 
the light. The subject of designing stained glass windows 
is taken up, and directions given for painting, firing, etching 
and embossing. The use of wooden mouldings or astragals 
to produce curved and interlaced forms. Details of French casements, etc. 





Draughtsmanship. —Edited by John Black. 94 
pages, illustrated by 75 engravings. 

This little manual is intended for those who desire some little 
knowledge of drawing, and to whom the study of the larger treatises 
would not be suitable. It briefly states about drawing instruments 
aud how to use them, geometry for draughtsman, the development 
of surfaces, perspective, together with some useful remarks about 
keeping records of builders’ work in a systematic manner, hints on 
the duties of the builder’s clerk. 

Enough practical information is given in this book to enable one 
to understand the principles of drawing as applied to the making of working plans and 
details for buildings. 

Wood Turning.— Edited by John Black. 91 pages, il¬ 
lustrated by 52 engravings. 

The art of wood turning is simply and clearly described in this volume, 
and will prove a useful manual for the amateur and young wood worker. 
After some introductory remarks about the history of wood turning, the 
following subjects are fully described and treated upon: Wood turners’ 
lathes, lathe details, mandrels, chucks of various kinds, tools, art of turning, 
turning classical columns, decorative side of wood turning, special lathes, 
slide rest, ornamental turning, examples and methods of making simple 
turned articles in the lathe, match boxes, reels,castors, candle stick, spice 
box, thimble holders, comb and brush tray, etc. 

Rafter and Brace Tables. —By H. J. Aurlie. 32 pages, illus¬ 
trated by 8 engravings. 

This time-saving reference book was prepared for the convenience of carpenters 
and builders, as it shows at sight the length of rafters and braces. As the tables are 
arranged in convenient form, their daily use will save time and mental effort and avoid 
the errors which are so liiiely to occur in finding of the lengths of any common hip, 
valley, or jack rafter. 

Besides these tables, there are others giving the decimal equivalents of parts of one 
inch for each sixty-fourth; strength of wooden posts; safe strength of bolts; weight 
and strength of chains and ropes; weights and measures, short cuts in mensuration, 
cost of chimneys, etc. 




Artificial Stone, Terra Cotter, etc.— 

Edited by John Black. 92 pages, illustrated 
by 29 engravings, 

The growing popularity of various forms of artificial 
stone for building purposes at the present time has led 
to the publication of this manual. It forms a compan¬ 
ion volume to CONCRETE, and concisely describes 
early manufacture and use of concrete and artificial 
stone; construction, manufacture and use of mosaic; 

S3 r stems of paving; artificial stone as a paving material; 
testing building stone; artificial sandstone sills, bricks 
and blocks; hydraulic machinery; hollow stone building 
blocks; scagli'ola and enameled slate: other methods of 
making artificial stone; terra cotta, its history, manufacture, cost, use, making a use¬ 
ful book of information about building materials that are constantly gaining favor. 

































DON’T CUT AND TRY 3 3 


But Lay Out Your Work Accurately 
by Up-to-Date Methods ^ ^ ^ ^ 





INC MADE EASY 


A Practical and Easily-Understood System of Laying Out and 
Framing Roofs Adapted to Modern Building Construction. 

The Methods are Made Clear by Nearly 100 Large & Clear Engravings 

By OWEN B. MAGINNIS, Architect 

Inspector of Buildings of the City of New York 

Author of “ How to Frame a House “ How to Measure Up Woodwork 
for Buildings ,” *“ Bricklaying," etc., etc. 


Second Edition, Revised and Greatly Enlarged, Now Ready 
^ *5 Over 160 Octavo Pages, Handsomely Bound in Cloth ^ ^ 

PRICE $ 1.00 PREPAID 


T HE carpenter or builder who will study the methods de¬ 
scribed in this book will realize the constructive value of 
every piece of timber which enters into a framed roof and will 
understand how to lay out every piece of timber used without 
wasting valuable time and material cutting and trying. 

The language used is that of the practical workman—scien¬ 
tific phrases and confusing terms have been avoided where 
possible—and everything has been made so plain that any one 
who will faithfully study the book will understand it from be¬ 
ginning to end. In fact, every problem in the book was “ tried*' 
on a boy who had had no experience in building work, and he 
understood every problem with a little study. This will show 
that the book is valuable to the beginner as well as the ad¬ 
vanced workman. 

Any intelligent mechanic will be able to save at least ten 
times the cost of this book in time and material during the first 
few weeks that he has it in use. 

INDUSTRIAL PUBLICATION CO. 

16 Thomas Street, N. Y. 


P. O. Box 1852 



















































































































































































































